# Whittaker functions for Steinberg representations of $GL(n)$ over a $p$-adic
field

Ehud Moshe Baruch and Markos Karameris

Let $G=GL_{n}(F)$ and let $(\pi_{St},V)$ be a (generalized) Steinberg
representation of $G$. It is well known that the space of Iwahori fixed
vectors in $V$ is one dimensional. The Iwahori Hecke algebra acts on this
space via a character. We determine the value of this character on a
particular Hecke algebra element and use this action to determine in full the
Whittaker function associated with an Iwahori fixed vector generalizing a
result of Baruch and Purkait for $GL_{2}(F)$. We show that the Iwahori fixed
vector is ”new” in the sense that it is not fixed by any larger parahoric. We
also show that the restriction of the (generalized) Steinberg representation
to $SL_{n}(F)$ remains irreducible hence we get the Whittaker function
attached to a Steinberg representation of $SL_{n}(F)$.

##  1 Introduction and Notations

In [6], Reeder considers two eigenvectors of the Iwahori Hecke algebra
$\mathcal{H}(G,J)$ corresponding to the spherical and the Steinberg
representation. The spherical Whittaker function is given by the well known
Casselman Shalika formula [7]. For a split reductive group, Reeder computes
the Whittaker function of the non-spherical vector on the diagonal values and
points that computing the values for a general element can be difficult. In
[5], Bump, Brubaker, Buciumas Henrik and Gustaffson give a formula that can be
used to compute the values of each summand $f_{w}$ that appears in the Iwahori
fixed vector. This however does not immediately provide a simple formula for
the Whittaker function in question. We work like [7] to give an explicit
formula for the Whittaker function of $\pi_{St}$ on $G$ and compute these
values for the case of the Steinberg fixed vector.

Let $F$ be a non-archimedean local field of zero characteristic and $|.|$ the
standard absolute value on $F$. We denote by $\mathfrak{p}$ the maximal ideal,
generated by the uniformizer $\varpi$, $\mathcal{O}_{F}$ the ring of integers
and $\mathfrak{f}=\mathcal{O}_{F}/\mathfrak{p}$ the residue field with $q$
elements. We also consider $G=GL_{n}(F)$ and the following subgroups inside
it: the center $\mathcal{Z}$ of $G$ of scalar matrices $z(t)$ with $t\in G$,
the maximal compact subgroup $K=GL_{n}(\mathcal{O}_{F})$, the standard Borel
subgroup $B$ of upper triangular, matrices in $G$, the maximal unipotent $N$
of upper triangular unipotent matrices, the restriction
$N_{\mathcal{O}_{F}}=N\cap K$, the maximal torus $T$ of diagonal matrices
$d(t_{1},...,t_{n})$ with entries $t_{i}\in F$ and also using the notation
$\varpi^{\bar{k}}=\varpi^{(k_{1},k_{2}\dots,k_{n})}=d(\varpi^{k_{1}},\dots,\varpi^{k_{n}})$
we define the set $T^{+}=\\{\varpi^{\bar{k}}|k_{1}\geq k_{2}\geq\dots
k_{n}\\}$, the Weyl group of permutation matrices $\mathbf{W}$ generated by
the simple reflections $s_{1},...,s_{n-1}$ and $w_{0}$ the longest Weyl
element and the Iwahori subgroup $J$ which is the preimage of the Borel in
$GL_{n}(\mathfrak{f})$ under the canonical projection induced by $K\to
GL_{n}(\mathfrak{f})$.

Denote by $\Phi=\\{\alpha_{i,j}|i\neq j\\}$ the associated root system of type
$A_{n-1}$ with positive simple roots $\Delta=\\{\alpha_{i,i+1},i=1\dots
n-1\\}$ where $\alpha_{i,j}:T\to F^{*}$ are given by
$\alpha_{i,j}(d(t_{1},t_{2},...,t_{n}))=t_{i}/t_{j}$. Let $e_{i,j}$ be the
$n\times n$ matrix with $1$ in the $(i,j)$ entry and zeroes every where else.
Then $x_{i,j}(t)=I+te_{i,j}$ are the one parameter groups corresponding to the
roots of $\Phi$. We also denote these as $x_{\alpha}(t)$ for $\alpha\in\Phi$.
Then the elements $d\in T$ act on the roots by conjugation as
$dx_{\alpha}(t)d^{-1}=x_{\alpha}(\alpha(d)t)$ and similarly for the Weyl
elements: $wx_{\alpha}(t)w^{-1}=x_{w\alpha}(t)$.

##  2 The Iwahori-Bruhat decomposition

Following Roger Howe [8] set: $s_{0}=\begin{pmatrix}0&\ldots&\varpi^{-1}\\\
\vdots&I_{n-2}&\vdots\\\ \varpi&\dots&0\end{pmatrix}$ and
$u=\begin{pmatrix}0&I_{n-1}\\\ \varpi&0\end{pmatrix}$

Let $\mathcal{H}(G,J)$ be the Hecke algebra of $J$ bi-invariant and compactly
supported functions on $G$. $\mathcal{H}(G,J)$ acts on spaces of $J$ invariant
vectors in a representation of $G$ and in particular on right $J$ invariant
functions on $G$. We denote by $X_{g}\in\mathcal{H}(G,J)$ the characteristic
function of $JgJ$. Then the left action of the Hecke algebra on a $J$
invariant function $F:G\to\mathbbm{C}$ is given by
$X_{g}(F)(x)=\int_{JgJ}F(xh)dh$ (see 1.2 of [3]). In fact we can compute this
action explicitly:

###### Lemma 1

For any $g\in G$ with coset decomposition
$JgJ=\overset{m}{\underset{i=1}{\bigcup}}\gamma_{i}J$ and $f$ a $J$ invariant
function. Then

$X_{g}(f)(x)=\sum\limits_{i=1}^{m}f(x\gamma_{i}).$

######  Theorem 1 (Iwahori, Matsumoto [1] Prop. 3.8)

$\mathcal{H}(G,J)$ is generated by $X_{s_{0}},X_{s_{i}},i\in\\{0,...,n-1\\}$
and $X_{u}$. The algebraic structure on $\mathcal{H}(G,J)$ is given by the
relations:
$1)(X_{s_{i}}-q)(X_{s_{i}}+1)=0\>,\;\;\;i=0,1,...,n-1\\\ 2)X_{u}^{n}=1,\\\
3)X_{u}X_{s_{i}}=X_{s_{i+1}}X_{u}\\\
4)X_{s_{i}}X_{s_{j}}X_{s_{i}}=X_{s_{j}}X_{s_{i}}X_{s_{j}},i\equiv\pm 1\mod
n\\\ 5)X_{s_{i}}X_{s_{j}}=X_{s_{j}}X_{s_{i}},i\not\equiv\pm 1\mod n$

The results on coset decompositions below are partially known from Iwahori-
Matsumoto [1]:

###### Lemma 2

For any $w\in\mathbf{W}$ with $\ell(w)=k$:

$JwJ=\underset{(t_{\alpha})\in(\mathcal{O}_{F}/\mathfrak{p})^{k}}{\bigcup}\prod\limits_{\alpha\in\Phi|w^{-1}\alpha\in\Phi^{-}}x_{\alpha}(t_{\alpha})wJ.$

Proof  By the Iwahori factorization for $J$:
$J=N_{\mathcal{O}_{F}}T^{0}N_{\mathfrak{p}^{-}}$ and
$N_{\mathfrak{p}}^{-}=\prod\limits_{\beta\in\Phi^{-}}x_{\beta}(\varpi
s_{\beta})$ where $N_{\mathcal{O}_{F}}=N\cap J$, $T^{0}=T\cap J$ and
$s_{\beta}\in\mathcal{O}_{F}$. But $w^{-1}T^{0}N_{\mathfrak{p}^{-}}w\subset
J$. It thus remains to check the terms in
$N_{\mathcal{O}_{F}}=\prod_{\alpha\in\Phi^{+}}x_{\alpha}(s_{\alpha})$. We now
have that $w^{-1}x_{\alpha}(s_{\alpha})w=x_{w^{-1}\alpha}(s_{\alpha})\in J$ if
and only if $w^{-1}\alpha\in\Phi^{+}$ or $w^{-1}\alpha\in\Phi^{-}$ and
$s_{\alpha}\in\mathfrak{p}$. Finally observing that
$\\#\\{\alpha\in\Phi|w^{-1}\alpha\in\Phi^{-}\\}=\ell(w^{-1})=\ell(w)$
completes the proof. $\Box$
Using Lemma 2, we obtain the refined Bruhat decomposition of $G$ relative to
the Iwahori subgroup:

######  Lemma 3 (Bruhat-Iwahori decomposition [1])

There exists a double coset decomposition of $G$ given by the following
disjoint union:

$G=\underset{d\in
T}{\bigcup}\underset{w\in\mathbf{W}}{\bigcup}N_{\mathcal{O}_{F}}dwJ$

.

##  3 Iwahori spherical vectors of Steinberg Representations

For a multiplicative character $\chi$ of $F^{*}$ we define a character
$\chi(det(g))$ for $g\in G$. We let $\delta_{B}$ be the modular character of
$B$ (so
$\delta_{B}(d(t_{1},t_{2},...,t_{n}))=|t_{1}|^{n-1}|t_{2}|^{n-3}...|t_{n}|^{1-n}$)
and denote by $I(\chi)=Ind_{B}^{G}(\delta_{B}^{1/2}\chi)$ the principal series
representation formed by normalized induction from $\chi$.

###### Definition 1

The generalized Steinberg representation corresponding to $\chi$ is the unique
irreducible quotient of $I(\chi\delta_{B}^{-1/2})$ which we denote by
$St^{\chi}$. If $\chi$ is unramified and trivial on $\mathcal{Z}$ then we call
this a Steinberg representation.

The Steinberg is the generic component of this principal series
representation. For $\chi=\chi\circ det$ as above $\chi^{w}=\chi$ meaning
$\chi\delta_{B}^{-1}$ is a regular character of $GL_{1}(F)$ i.e.
$\chi\delta_{B}^{-1}=(\chi\delta_{B}^{-1})^{w}\iff w=1$.
The Jacquet functor $V\to(V)_{N}$ is an exact functor from the category of
smooth representations of $G$ to the category of smooth representations of the
maximal torus $T$ and can be computed explicitly using a trivial modification
of Lemma 8.1.2 of Casselman [2]:

###### Proposition 1

For any character $\chi$ as above we have that
$(St^{\chi})_{N}\simeq\chi\delta_{B}$.

###### Definition 2

An irreducible admissible representation $(\pi,V)$ of $G$ is called Iwahori
spherical if it has a non-zero vector fixed by the Iwahori subgroup $J$. In
this case the space of fixed vectors $V^{J}$ is a simple module for the
Iwahori Hecke algebra $\mathcal{H}(G,J)$.

Assuming $\tau^{n}=1$ forces $\tau(\varpi^{k})=\epsilon^{k}$ for some $n$-th
root of unity $\epsilon$. In this case Proposition 2.1 of [10] gives us the
Casselman basis of $I(\tau\delta_{B}^{1/2})^{J}$ which consists of
characteristic functions of the form:

$f_{w}(g)=\begin{cases}\tau(b)\delta_{B}(b)&\text{if }g=bwj,b\in B,j\in J,\\\
0&\text{otherwise}\end{cases}.$

We write again the Iwahori factorization of the Iwahori subgroup
$J=N_{\mathcal{O}_{F}}T^{0}N^{-}_{\mathfrak{p}}$. Then we have the following
result due to Borel [3] which can also be found in more detail in Casselman’s
notes [4]:

###### Proposition 2

For an admissible irreducible representation $(\pi,V)$ of $G$ the canonical
projection $V\to(V)_{N}$ induces an isomorphism $V^{J}\to(V)_{N}^{T^{0}}$.

###### Corollary 1

By the above proposition and Proposition 1, and since $\delta_{B}$ and $\tau$
are unramified and thus $T^{0}$ invariant, it follows that
$dim((St^{\tau})_{N}^{T^{0}})=1$ meaning $dim((St^{\tau})^{J})=1$.

It is well known that the generators $X_{s_{i}}$ act on $(St^{\tau})^{J}$ with
the scalar $-1$. We will give an alternative proof and also obtain the action
of the generator $X_{u}$.

###### Proposition 3

(p.23 of [6]) The Iwahori fixed part of the principal series representation
$I(\tau\delta_{B}^{1/2})^{J}$ contains up to scalar multiplication exactly two
eigenvectors of $\mathcal{H}(G,J)$ which are given by the formulas
$\phi^{-}(g)=\sum\limits_{w\in W}(-q)^{-\ell(w)}f_{w}(g)$ and
$\phi^{+}(g)=\sum\limits_{w\in W}f_{w}(g)$.

Proof  There are only two complex characters of the affine part of
$\mathcal{H}(G,J)$ given by $X_{s_{\alpha}}\to-1,\forall\alpha\in\Delta$ and
$X_{s_{\alpha}}\to q,\forall\alpha\in\Delta$. For these two characters we
identify the corresponding fixed vectors by using induction on the order of an
element of $W$. Let $\phi^{-}(g)=\sum\limits_{w\in W}\lambda_{w}f_{w}(g)$ be
such that $T_{\alpha}(\phi^{-})(g)=-\phi^{-}(g)$, then trivially
$\lambda_{1}=\lambda_{1}(-q)^{-\ell(1)}$. Assume that for all $w_{1}\in W$
with $\ell(w)=n$ we have that $\lambda_{w_{1}}=(-q)^{-n}\lambda_{1}$. For
every $w_{2}\in W$ with $\ell(w_{2})=n+1$ write $w_{2}=w_{1}s_{\alpha}$ for
some $s_{\alpha}\in\Delta$ so that $\ell(w_{1})=n$. Then from Lemma 1, for any
$w\in W$:

$X_{s_{i}}(f_{w})(w_{1})=\sum\limits_{t\in\mathcal{O}_{F}/\mathfrak{p}}f_{w}(w_{1}x_{\alpha}(t)s_{\alpha})=\sum\limits_{t\in\mathcal{O}_{F}/\mathfrak{p}}f_{w}(x_{w_{1}\alpha}(t)w_{1}s_{\alpha}).$

Observe however that $x_{w_{1}\alpha}(t)\in N_{\mathcal{O}_{F}}$ since
$\ell(w_{1}s_{\alpha})>\ell(w_{1})\iff w_{1}\alpha\in\Phi^{+}$. This implies
that $X_{s_{\alpha}}(f_{w})(w_{1})\neq 0$ if and only if
$w=w_{1}s_{\alpha}=w_{2}$ and in that case:

$X_{s_{\alpha}}(f_{w_{2}})(w_{1})=q.$

But then
$-\phi^{-}(w_{1})=X_{s_{\alpha}}(\phi^{-})(w_{1})\implies-\lambda_{w_{1}}=q\lambda_{w_{2}}$
which by the induction hypothesis implies
$\lambda_{w_{2}}=(-q)^{-1}(-q)^{-n}\lambda_{1}=(-q)^{-(n+1)}\lambda_{1}$.
The same proof in the case of $\phi^{+}$ with
$T_{\alpha}(\phi^{+})(g)=-\phi^{+}(g)$ means changing the inductive hypothesis
to $\ell(w_{1})=\lambda_{1}$ and $X_{s_{\alpha}}(\phi^{+})(g)=q\phi^{+}(g)$.
Then we get in the same way
$q\lambda_{w_{1}}=q\lambda_{2}\implies\lambda_{w_{2}}=\lambda_{1}$ completing
the proof. $\Box$

Note that $\phi^{+}$ is a $K$ spherical function and thus is not inside the
Steinberg subrepresentation. In the next Proposition we will obtain the action
of the generators of the Hecke Algebra on $\phi^{-}$. This action is well
known for the simple reflections $X_{s_{i}}$.

###### Proposition 4

The Hecke algebra character of the twisted Steinberg $St^{\tau}$ denoted
$\rho_{\tau}:\mathcal{H}(G,J)\to\mathbbm{C}$ and corresponding to the
unramified character
$\tau(\varpi^{\bar{k}})=\epsilon^{\sum\limits_{i=1}^{n}k_{i}}$ is given by

$\rho_{\tau}(X_{s_{i}})=-1$

for all simple reflections and

$\rho_{\tau}(X_{u})=(-1)^{n-1}\epsilon.$

Proof  It is sufficient to compute the action of the affine part of
$\mathcal{H}(G,J)$. We thus observe that
$u=\varpi^{(0,\dots,0,1)}s_{n-1}...s_{1}$ we set $u^{\prime}=s_{n-1}...s_{1}$
and $d_{u^{\prime}}=\varpi^{(0,\dots,0,1)}$. From Proposition 3 we have that
$T_{u}(\phi)(g)=\sum\limits_{w\in W}(-q)^{-\ell(w)}T_{u}(f_{w})(g)$. For $g=1$
then, from Lemma 1 $T_{u}(f_{w})(1)=f_{w}(u)=f_{w}(d_{u^{\prime}}u^{\prime})$.
This means that $T_{u}(f_{w})(1)=0$ unless $w=u^{\prime}$ in which case:

$T_{u}(f_{u^{\prime}})(1)=f_{u^{\prime}}(d_{u^{\prime}}u^{\prime})=\delta_{B}(d_{u^{\prime}})\tau(d_{u^{\prime}})=q^{n-1}\epsilon.$

This readily implies now that:

$T_{u}(\phi)(1)=(-q)^{-\ell(u^{\prime})}q^{n-1}=(-q)^{-(n-1)}q^{n-1}\epsilon=(-1)^{n-1}\epsilon.$

$\Box$

##  4 Iwahori fixed Whittaker functions for $GL(n)$

We compute the Whittaker function associated to a Steinberg representation.
For an unramified character $\psi:F\to\mathbbm{C}^{\times}$ we extend it to a
character $\psi:N\rightarrow\mathbbm{C}^{\times}$ by
$\psi(n)=\sum\psi(n_{i,i+1})$. Let $\mathcal{W}(\pi_{St^{\tau}},\psi)$ be the
Whittaker model of $St^{\tau}$ with character $\psi$ then
$\mathcal{W}(\pi_{St^{\tau}},\psi)\cong St^{\tau}$. We are interested in
$\mathcal{W}(\pi_{St^{\tau}},\psi)^{J}$ which by Corollary 1 has dimension
$1$. These generalised Steinberg representations correspond to twists of the
sign representation of $\mathcal{H}(G,J)$ by a root of unity
$\epsilon\in\mathbbm{C}^{\times}$ as showed in Proposition 4. We can thus
denote such a twist of the Steinberg by $St^{\epsilon}$ instead of
$St^{\tau}$. Let $W\in\mathcal{W}(\pi_{St^{\tau}},\psi)^{J}$ with $W\not\equiv
0$, then $W(gj)=W(g),\forall g\in G$ and $j\in J$ and
$(F*W)(g)=\rho_{\tau}(F)W(g)$ with $\rho_{\tau}$ the character of
$\mathcal{H}(G,J)$ as above. From Lemma 3 it is clear that the values of $W$
are completely determined by the elements in $TW$ and specifically if
$d=d(t_{1},...,t_{n})$ then $W(dw)=W(d(|t_{1}|,...,|t_{n}|)w)$ since the units
can be conjugated inside $J$. This means that we can restrict ousrelves to
only computing the value of $W$ on $\varpi^{\bar{k}}w$. In what follows we
always assume $W$ is as above.
Define $\langle\alpha_{i,j},\bar{k}\rangle=k_{i}-k_{j}$. A generalization of
the set $T^{+}$ is the following result a variant of which appears in [5]:

###### Lemma 4

Let $d=\varpi^{(k_{1},...,k_{n-1},k_{n})}=\varpi^{\bar{k}}$, then $W(dw)=0$
unless $\langle\alpha,\bar{k}\rangle\geq\begin{cases}0&\text{if
$w^{-1}\alpha\in\Phi^{+}$}\\\ -1&\text{if
$w^{-1}\alpha\in\Phi^{-}$}\end{cases}$ for all $\alpha\in\Delta$. A weight
$\bar{k}$ satisfying this condition will be called $w$-dominant.

Proof  Let $s\in F$ and $\alpha\in\Delta$ then:
$\psi(x_{\alpha}(s))W(dw)=W(x_{\alpha}(s)dw)=W(dd^{-1}x_{\alpha}(s)dw)=W(dww^{-1}x_{\alpha}(\varpi^{-\langle\alpha,\bar{k}\rangle}s)w)=W(dwx_{w^{-1}\alpha}(\varpi^{-\langle\alpha,\bar{k}\rangle}s))$.
If $x_{w^{-1}\alpha}(\varpi^{-\langle\alpha,\bar{k}\rangle}s)\in J$, then we
have
$\psi(x_{\alpha}(s))W(dw)=W(dwx_{w^{-1}\alpha}(\varpi^{-\langle\alpha,\bar{k}\rangle}s))=W(dw)$
and thus picking an $s\in F$ with $\psi(x_{\alpha}(s))\neq 1$ we have
$W(dw)=0$. On the other hand $W(dw)\neq 0$ implies
$\varpi^{-\langle\alpha,\bar{k}\rangle}s\not\in\mathfrak{p}$ or
$\varpi^{-\langle\alpha,\bar{k}\rangle}s\not\in\mathcal{O}_{F}$ depending on
whether $w^{-1}\alpha\in\Phi^{-}$ or $w^{-1}\alpha\in\Phi^{+}$. Noting that we
can pick $s\in F:|s|=q$ completes the proof. $\Box$

###### Remark 1

The above condition can be translated as follows: $W(dw)=0$ except when
$k_{i}\geq k_{i+1}$ if $w^{-1}$ has an ascend at the place $i$ or $k_{i}\geq
k_{i+1}-1$ if $w^{-1}$ has an descend at the place $i$. Thus $W(dw)\neq 0$ if
and only if $k_{1}-g_{1}\geq k_{2}-g_{2}\geq...\geq k_{n}-g_{n}$ where
$g_{i}=\\#\\{j\geq i:w^{-1}(j)>w^{-1}(j+1)\\}$.

###### Corollary 2

For any $d\in T$ we have $W(dw)=0$ unless $d\in T_{w}^{+}T^{0}$.

###### Definition 3

We denote the semigroup of $w$-dominant diagonal matrices by $T_{w}^{+}$ and
define the element $d_{w}=\\{\varpi^{\bar{k}}\in T_{w}^{+}:k_{n}=0\text{ and
}\langle\alpha,\bar{k}\rangle=\begin{cases}0&\text{if
$w^{-1}\alpha\in\Phi^{+}$}\\\ -1&\text{if
$w^{-1}\alpha\in\Phi^{-}$}\end{cases},\forall\alpha\in\Delta\\}$.

The element $d_{w}$ can be realized as the unique maximal valuation element of
$T_{w}^{+}/\mathcal{Z}$. It is easy to see that $T_{w}^{+}=d_{w}T^{+}$, so in
particular $T^{+}\subseteq T_{w}^{+},\forall w\in\mathbf{W}$. The following
follows from Lemma 1:

###### Lemma 5

For a Weyl element $w$ and $d\in T^{+}$ the following relation is satisfied:

$W(dw)=(-q)^{-\ell(w)}W(d).$

Proof  We use the relation $X_{a}*X_{b}=X_{ab}$ if $\ell(a)+\ell(b)=\ell(ab)$.
Using an inductive argument we easily see that:
$X_{w}(W)(g)=(-1)^{\ell(w)}W(g)$. From Lemma 2: $JwJ=\underset{(t_{1},\dots
t_{\ell(w)})\in\mathcal{(}{O}_{F}/\mathfrak{p})^{\ell(w)}}{\bigcup}\prod\limits_{j=1}^{\ell(w)}x_{\alpha_{j}}(t_{j})wJ$
with $\prod\limits_{j=1}^{\ell(w)}x_{\alpha_{j}}(t_{j})\in
N_{\mathcal{O}_{F}}$ and $\mu(JwJ)=|JwJ/J|=q^{\ell(w)}$. From Lemma 1 now
$(-1)^{\ell(w)}W(d)=\sum\limits_{(t_{1},\dots
t_{\ell(w)})\in\mathcal{(}{O}_{F}/\mathfrak{p})^{\ell(w)}}W(d\prod\limits_{j=1}^{\ell(w)}x_{\alpha_{j}}(t_{j})w)=q^{\ell(w)}W(dw)\implies
W(dw)=(-q)^{-\ell(w)}W(d)$. $\Box$

######  Theorem 2 (Diagonal Whittaker values)

Let $W\in\mathcal{W}(\pi,\psi)^{J}$ with $W(1)=1$ be such that

$W(zg)=W(g),\;z\in Z,g\in G$

and that

$X_{s_{i}}(W)(g)=-W(g),\;\;\;X_{u}(W)(g)=\epsilon W(g)$

Then $W$ has the following diagonal values:

$W(\varpi^{(k_{1},k_{2},\dots,k_{n})})=\begin{cases}\epsilon^{\sum\limits_{i=1}^{n}k_{i}}(-1)^{(n-1)\sum\limits_{i=1}^{n}k_{i}}\delta_{B}(d)&\text{if
$\varpi^{(k_{1},k_{2},\dots,k_{n})}\in T^{+}$}\\\
0&\text{otherwise}\end{cases}.$

Proof  We note again that

$u=\varpi^{(0,0,...,0,1)}s_{n-1}s_{n-2}...s_{1}$

We also have that $s_{i}^{2}=1$ and :

$s_{i}\varpi^{\underset{i+1\text{-th
place}}{(0,0,\dots,1,\dots,0)}}s_{i}=\varpi^{\underset{i\text{-th
place}}{(0,0,\dots,1,\dots,0)}},i\in\\{1,...,n-1\\}.$

This gives us the following $n$ relations:

$u=\varpi^{(0,0,\dots,0,1)}s_{n-1}s_{n-2}...s_{1}$
$s_{n-1}u=\varpi^{(0,...,1,0)}s_{n-2}...s_{1}$ $\vdots$
$s_{1}s_{2}...s_{n-1}u=\varpi^{(1,0,...,0,0)}$

Since $X_{u}(W)(g)=W(gu)$ it follows that $W(gu)=\epsilon W(g)$ for all $g\in
G$. Using this relation and setting each time $g=ds_{i}...s_{n-1}$ with
$d=d(\varpi^{k_{1}},\dots,\varpi^{k_{n-1}})$, where $d\in T^{+}$ i.e.
$k_{1}\geq k_{2}\geq...\geq k_{n-1}\geq k_{n}$ we obtain the relations (*):

$\epsilon W(d)=W(d\varpi^{(0,0,\dots,0,1)}s_{n-1}s_{n-2}...s_{1})$ $\epsilon
W(ds_{n-1})=W(d\varpi^{(0,...,0,1,0)}s_{n-2}...s_{1})$ $\vdots$ $\epsilon
W(ds_{2}...s_{n-1})=W(d\varpi^{(0,1,...,0,0)}s_{1})$

From the last relation, using Lemma 5 on both sides of (*) and noting that
$\ell(s_{j}...s_{1})=\ell(s_{1}...s_{j})=j$ we see that:

$W(\varpi^{(k_{1},k_{2},...,k_{n}}))=\epsilon(-q)^{(n-3)}W(\varpi^{(k_{1},k_{2}+1,...,k_{n}})).$

and iterating we get:

$W(\varpi^{(k_{1},k_{2},k_{3},...,k_{n}}))=\epsilon^{k_{2}-k_{1}}(-q)^{(n-3)(k_{1}-k_{2})}W(\varpi^{(k_{1},k_{1},k_{3},...,k_{n}})).$

Next we use the second to last relation and similarly get:

$W(\varpi^{(k_{1},k_{1},k_{3},k_{4},...,k_{n}}))=\epsilon^{k_{3}-k_{1}}(-q)^{(n-5)(k_{1}-k_{3})}W(\varpi^{(k_{1},k_{1},k_{1},k_{4},...,k_{n}})).$

Continuing in this manner we finally obtain:

$W(d)=\prod\limits_{i=1}^{n}\epsilon^{k_{i}-k_{1}}(-q)^{(n+1-2i)(k_{1}-k_{i})}W(\varpi^{(k_{1},k_{1},k_{2},\dots,k_{1})})=\prod\limits_{i=1}^{n}\epsilon^{k_{i}-k_{1}}(-q)^{(n+1-2i)(k_{1}-k_{i})}.$

The result now follows by observing that
$\sum\limits_{i=1}^{n}(2i-1-n)k_{1}=0$ and that
$q^{-\sum\limits_{i=1}^{n}(1+n-2i)k_{i}}=\delta_{B}(d)$. $\Box$

###### Lemma 6

For any Weyl element $w$:
$Jd_{w}wJ=\underset{\bar{s}=(s_{\alpha})}{\bigcup}\prod\limits_{\alpha\in\Phi^{-}}x_{\alpha}(\varpi
s_{\alpha})d_{w}wJ$ where
$s_{\alpha}\in\mathcal{O}_{F}/\mathfrak{p}^{n_{\alpha}}$ for some
$n_{\alpha}\in\mathbbm{N}\cup\\{0\\}$ and each $\alpha\in\Phi^{-}$.

Proof  We write $J=N^{-}_{\mathfrak{p}}T^{0}N_{\mathcal{O}_{F}}$ and
$N_{\mathcal{O}_{F}}=\prod\limits_{\beta\in\Phi^{+}}x_{\beta}(s_{\beta})$. We
will show that
$x_{w^{-1}(\beta)}{(\varpi^{\langle\beta,d_{w}^{-1}\rangle}s_{\beta})}\in
J,\forall\beta\in\Phi^{+}$ .It is sufficient to prove the statement for
positive simple roots. By the definition of $d_{w}$ however it is
straightforward to check that
$x_{w^{-1}(\beta)}{(\varpi^{\langle\beta,d_{w}^{-1}\rangle}s_{\beta})}\in
J,\forall\beta\in\Delta^{+}$ and $s_{\beta}\in\mathcal{O}_{F}$. $\Box$

###### Lemma 7

For every Weyl element $w\in\mathbf{W}$ we have
$d_{w_{0}w}=w_{0}d_{w}w_{0}d_{w_{0}}z_{w}$ for some $z_{w}\in\mathcal{Z}$.

Proof  We observe that $w^{-1}(i)>w^{-1}(i+1)\iff
w^{-1}w_{0}(n+1-i)>w^{-1}w_{0}(n+1-(i+1))$ thus we have that if
$d_{w}=(\varpi^{-g_{1}},...,\varpi^{-g_{n}})$ then the exponents
$g^{\prime}_{i}$ of $d_{w_{0}w}$ which are the number of descents of
$(w_{0}w)^{-1}$ occuring after place $i$ are given by
$g^{\prime}_{i}=n-i-(g_{1}-g_{n-i+1})$. This imples
$d_{w_{0}w}=(\varpi^{-(n-1)-g_{n}+g_{1}},...,\varpi^{-(n-i)-g_{n-i+1}+g_{1}},...)$
which is easily seen to be $w_{0}d_{w}w_{0}d_{w_{0}}z(\varpi^{g_{1}})$. $\Box$

######  Theorem 3 (Whittaker values at every cell $Tw$)

Let as before $d=\varpi^{\bar{k}}$, then under the assumptions of Theorem 2

$W(dw)=\begin{cases}\epsilon^{\sum\limits_{i=1}^{n}k_{i}}(-1)^{(n-1)\sum\limits_{i=1}^{n}k_{i}}\delta_{B}(d)(-q)^{-\ell(w)}&\text{if
$\bar{k}$ is $w$-dominant}\\\ 0&\text{otherwise}\end{cases}$

Proof  Since every element of the Iwahori Hecke algerba acts as scalar
multiplication, $X_{d_{w}w}(W)(g)=\lambda_{w}W(g)$ for some
$\lambda_{w}\in\mathbbm{C}$. By Lemma 6 we have
$\lambda_{w}W(g)=\sum\limits_{\bar{s}}W(g\prod\limits_{\alpha\in\Phi^{-}}x_{\alpha}(\varpi
s_{\alpha})d_{w}w)$. Setting $g=dd_{w_{0}}w_{0}(=dw_{0}d_{w_{0}}^{-1})$ we
obtain $w_{0}d_{w_{0}}^{-1}\prod\limits_{\alpha\in\Delta^{-}}x_{\alpha}(\varpi
s_{\alpha})d_{w_{0}}w_{0}=\prod\limits_{\alpha\in\Delta^{-}}x_{w_{0}(\alpha)}(\varpi
s_{\alpha}\varpi^{\langle\alpha,d_{w_{0}}^{-1}\rangle})\in
N_{\mathcal{O}_{F}}$ ,since $\langle\alpha,d_{w_{0}}^{-1}\rangle={-1}$ and
$\psi(dx_{w_{0}(\alpha)}(\varpi
s_{\alpha}\varpi^{\langle\alpha,d_{w_{0}}^{-1}\rangle})d^{-1})=1$ for any
$\alpha\in\Phi^{-}\backslash\Delta^{-}$ and $d\in T$. Thus we can conjugate
$\prod\limits_{\alpha\in\Phi^{-}}x_{w_{0}(\alpha)}(\varpi
s_{\alpha}\varpi^{\langle\alpha,d_{w_{0}}^{-1}\rangle})$ with any $d\in
T^{+}$. We now get for any $d\in T^{+}$:
$\lambda_{w}W(dd_{w_{0}}w_{0})=\mu(Jd_{w}wJ)W(dd_{w_{0}}w_{0}d_{w}w_{0}w_{0}w)$.
From Lemma 7 now this becomes:

$\lambda_{w}W(dd_{w_{0}}w_{0})=\mu(Jd_{w}wJ)W(dd_{w_{0}w}w_{0}w),\forall d\in
T.$

Setting $w=w_{0}$ and comparing scalar terms with Lemma 5 for
$d=d_{w_{0}}^{-1}$ we get that
$\lambda_{w_{0}}=(-q)^{\ell(w_{0})}\mu(Jd_{w_{0}}w_{0}J)W(d_{w_{0}}^{-1})$.
Substituting the value of $\lambda_{w_{0}}$ and setting $d\to dd_{w_{0}}^{-1}$
leads to:

$W(dw_{0})=(-q)^{-\ell(w_{0})}W(d_{w_{0}}^{-1})^{-1}W(dd_{w_{0}}^{-1}),\forall
d\in T$

Similarly we set $w\to w_{0}w$ and compute
$\lambda_{w_{0}w}=(-q)^{\ell(w_{0})-\ell(w)}\mu(Jd_{ww_{0}}ww_{0}J)W(d_{w_{0}}^{-1}d_{w})$.
Substituting this value of $\lambda_{w_{0}w}$ and using the above relation for
the $W(dw_{0})$ term implies:

$W(dw)=(-q)^{-\ell(w)}W(d_{w}^{-1})^{-1}W(dd_{w}^{-1}),\forall d\in T\;\;(**)$

which allows us to compute $W(dw)$ for any $w\in\mathbf{W}$ and $d\in T$ since
$W$ is multiplicative on $T^{+}$. $\Box$

###### Corollary 3

If $W\in\mathcal{W}(\pi_{St^{\tau}},\psi)^{J}$ and $W(1)=0$ then relations (*)
imply that $W(d)=0,\forall d\in T^{+}\iff W(d)=0,\forall d\in T$. Relation
(**) then implies that $W(dw)\equiv 0$ in every cell thus contradicting the
assumption $W\not\equiv 0$. This strengthens a more general non-vanishing
result in [6].

##  5 Whittaker Functions and New-vectors

From the relation $(F*W)(g)=\rho_{\tau}(F)W(g)$ and the uniqueness in Theorem
3 we conclude that:

###### Theorem 4

The function described in Theorem 3

$W(dw)=\begin{cases}(\tau\delta_{B})(d)(-q)^{-\ell(w)}&\text{if $\bar{k}$ is
$w$-dominant}\\\ 0&\text{otherwise}\end{cases}$

with $d=\varpi^{\bar{k}}$, is the unique Whittaker function $W_{v}$ that
corresponds to the unique Iwahori spherical vector $v\in(St^{\tau})^{J}$.

We now use this explicit description of the Whittaker function associated to
$v\in St^{\epsilon}$ to show that $v$ is not fixed by any other parahoric
subgroup of $G$ for all generalized Steinberg representations.
For $S=\\{s_{i_{1}},...,s_{i_{r}}\\}$ a subset of the simple permutations
generating $\mathbf{W}$, let $\mathbf{W}_{S}$ be the subset of $\mathbf{W}$
generated by the elements of $S$.

###### Theorem 5

For any parahoric $K_{S}=J\mathbf{W}_{S}J$ with $S\neq\\{1\\}$ it holds that
$(St^{\epsilon})^{K_{S}}=\\{0\\}$.

Proof  Assume $v\in(St^{\epsilon})^{K_{S}}$ is non-zero, then
$v\in(St^{\epsilon})^{J}$ and thus $W_{v}$ is of the form we compute in
Theorem 3. Then $v\in(St^{\epsilon})^{K_{S}}\implies
W_{v}(gk_{S})=W(g),\forall k_{S}\in K_{S}$ so picking $k_{S}=s_{i}$ for some
$s_{i}\in S$ we see that $W(gs_{i})=W(g),\forall g\in G$. Setting
$g=d_{s_{i}}$ yields: $W(d_{s_{i}}s_{i})=W(d_{s_{i}})=0$ since
$d_{s_{i}}\not\in T^{+}$. This however contradicts the formula in Theorem 3
according to which $W(d_{s_{i}}s_{i})\neq 0$. $\Box$

##  6 Whittaker Functions for $SL(n)$

We will use basic notions from $l$-indistinguishability to describe the
Iwahori spherical Whittaker functions of $SL_{n}(F)$ in the same way. Let
$G^{\prime}=SL_{n}(F)\times\mathcal{Z}$. The representations of $G^{\prime}$
are trivially in bijection with the representations of $SL_{n}(F)$ up to twist
by a fixed central character of $G^{\prime}$.

###### Lemma 8

$St^{\tau}|_{SL_{n}(F)}$ remains irreducible as a representation of
$SL_{n}(F)$.

Proof  Let $\chi$ be a character of $G$ that is trivial on $G^{\prime}$. From
the exactness of the Jacquet functor we have that if $\chi\otimes
St^{\tau}\simeq St^{\tau}$ then $(\chi\otimes
St^{\tau})_{N}\simeq(St^{\tau})_{N}$ and thus since $\chi\otimes I(\tau)\simeq
I(\chi\tau)$ and $\chi\tau\delta_{B}$ is a regular character, this implies by
Proposition 1 that $\chi\tau\delta_{B}\simeq\tau\delta_{B}$. Since these are
characters though this means
$\chi\tau\delta_{B}=\tau\delta_{B}\implies\chi\equiv 1$. From Lemma 2.1 d) of
[9] this means $St^{\tau}$ is irreducible as a representation of $G^{\prime}$
and thus of $SL_{n}(F)$. $\Box$

###### Theorem 6

The restriction of the Whittaker function described in Theorem 4 to
$SL_{n}(F)$ (i.e. for $d\in T:det(d)=1$) is the unique up to scalar
multiplication Iwahori spherical Whittaker function of $SL_{n}(F)$
corresponding to $(St^{\tau})^{J}$.

Notice that we get only one Whittaker function for all generalized Steinberg
representations $(St^{\tau})^{J}$. This is because in $SL_{n}(F)$ all of them
restrict to the same irreducible representation since $\tau\otimes St\simeq
St^{\tau}$ by definition and $\tau$ is trivial on $G^{\prime}$. In Hecke
algebra terms this can be observed by restricting the action of
$\mathcal{H}(G,J)$ to the action of $X_{s_{i}}$ only.

## References

  * [1] N. Iwahori, H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of $p$-adic Chevalley groups, Publications Mathématiques de l’I.H.É.S., tome 25 (1965), p. 5-48
  * [2] W. Casselman, The Steinberg character as a true character, in Moore, Calvin C. (ed.), Harmonic analysis on homogeneous spaces (Williams Coll., Williamstown, Mass., 1972), Proc. Sympos. Pure Math., vol. XXVI, Providence, R.I.: American Mathematical Society, 413–417, ISBN 978-0-8218-1426-0
  * [3] A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Inv. Math., 35, 233- 259 (1976)
  * [4] W. Casselman, An introduction to the theory of admissible representations of reductive p-adic groups, Unpublished notes
  * [5] B. Brubaker, V. Buciumas, D. Bump, Henrik P. A. Gustafsson, Colored Vertex Models and Iwahori Whittaker Functions, 2019. .arXiv: 1906.04140 [math.RT]
  * [6] M. Reeder, $p$-adic Whittaker functions and vector bundles on flag manifolds, Comp. Math. 85, No 1 (1980), p. 9 - 36
  * [7] T. Shintani, On an Explicit Formula for Class-$1$ ”Whittaker Functions” on $GL_{n}$ over $p$-adic Fields, Proc. Japan Acad. 52, (1976), p. 180 - 182
  * [8] R. Howe, Affine-like Hecke algebras and p-adic representation theory in Iwahori-Hecke Algebras and their Representation Theory, Lecture Notes in Mathematics 1804 (2002), 27–69.
  * [9] S.S. Gelbart, A. W. Knapp, L-indistinguishability and R Groups for the Special Linear Group, Adv. Math. 43 (1982), p. 101-121
  * [10] W. Casselman, The unramified principal series of p-adic groups. I. The spherical function, Comp. Math. 40, No 3 (1980), p. 387-406


# SEGA: Instructing Diffusion using Semantic Dimensions

Manuel Brack1,2 Felix Friedrich2,3 Dominik Hintersdorf2 Lukas Struppek2
Patrick Schramowski1,2,3,4 Kristian Kersting1,2,3,5
1German Research Center for Artificial Intelligence (DFKI),
2Computer Science Department, TU Darmstadt 3Hessian.AI,
4LAION, 5Centre for Cognitive Science, TU Darmstadt
<EMAIL_ADDRESS>
# SEGA: Instructing Any Text-to-Image Model
using Semantic Dimensions

Manuel Brack1,2 Felix Friedrich2,3 Dominik Hintersdorf2 Lukas Struppek2
Patrick Schramowski1,2,3,4 Kristian Kersting1,2,3,5
1German Research Center for Artificial Intelligence (DFKI),
2Computer Science Department, TU Darmstadt 3Hessian.AI,
4LAION, 5Centre for Cognitive Science, TU Darmstadt
<EMAIL_ADDRESS>
# SEGA: Instructing Text-to-Image Models
using Semantic Guidance

Manuel Brack1,2 Felix Friedrich2,3 Dominik Hintersdorf2 Lukas Struppek2
Patrick Schramowski1,2,3,4 Kristian Kersting1,2,3,5
1German Research Center for Artificial Intelligence (DFKI),
2Computer Science Department, TU Darmstadt 3Hessian.AI,
4LAION, 5Centre for Cognitive Science, TU Darmstadt
<EMAIL_ADDRESS>
###### Abstract

Text-to-image diffusion models have recently received a lot of interest for
their astonishing ability to produce high-fidelity images from text only.
However, achieving one-shot generation that aligns with the user’s intent is
nearly impossible, yet small changes to the input prompt often result in very
different images. This leaves the user with little semantic control. To put
the user in control, we show how to interact with the diffusion process to
flexibly steer it along semantic directions.

This semantic guidance (Sega) generalizes to any generative architecture using
classifier-free guidance. More importantly, it allows for subtle and extensive
edits, changes in composition and style, as well as optimizing the overall
artistic conception. We demonstrate Sega’s effectiveness on both latent and
pixel-based diffusion models such as Stable Diffusion, Paella and DeepFloyd-IF
using a variety of tasks, thus providing strong evidence for its versatility,
flexibility and improvements over existing methods111Implementation available
in diffusers:
https://huggingface.co/docs/diffusers/api/pipelines/semantic_stable_diffusion.

##  1 Introduction

The recent popularity of text-to-image diffusion models (DMs)
[saharia2022photorealistic, ramesh2022hierarchical, rombach2022High] can
largely be attributed to their versatility, expressiveness, and—most
importantly—the intuitive interface they provide to users. The generation’s
intent can easily be expressed in natural language, with the model producing
faithful interpretations of a text prompt. Despite the impressive capabilities
of these models, the initially generated images are rarely of high quality.
Accordingly, a human user will likely be unsatisfied with certain aspects of
the initial image, which they will attempt to improve over multiple
iterations. Unfortunately, the diffusion process is rather fragile as small
changes to the input prompt lead to entirely different images. Consequently,
fine-grained semantic control over the generation process is necessary, which
should be as easy and versatile to use as the initial generation.

Previous attempts to influence dedicated concepts during the generation
process require additional segmentation masks, extensions to the architecture,
model fine-tuning, or embedding optimization [avrahmi2022blended,
hertz2022prompt, kawar2022Imagic, wu2022uncovering]. While these techniques
produce satisfactory results, they disrupt the fast, exploratory workflow that
is the strong suit of diffusion models in the first place. We propose Semantic
Guidance (Sega) to uncover and interact with semantic directions inherent to
the model. Sega requires no additional training, no extensions to the
architecture, nor external guidance and is calculated within a single forward
pass. We demonstrate that this semantic control can be inferred from simple
textual descriptions using the model’s noise estimate alone. With this, we
also refute previous research claiming these estimates to be unsuitable for
semantic control [kwon2022diffusion]. The guidance directions uncovered with
Sega are robust, scale monotonically, and are largely isolated. This enables
simultaneous applications of subtle edits to images, changes in composition
and style, as well as optimizing the artistic conception. Furthermore, Sega
allows for probing the latent space of diffusion models to gain insights into
how abstract concepts are represented by the model and how their
interpretation reflects on the generated image. Additionally, Sega is
architecture-agnostic and compatible with various generative models, including
latent[rombach2022High, rampas2022fast] and pixel-based
diffusion[saharia2022photorealistic].

In this paper, we establish the methodical benefits of Sega and demonstrate
that this intuitive, lightweight approach offers sophisticated semantic
control over image generations. Specifically, we contribute by (i) devising a
formal definition of Semantic Guidance and discussing the numerical intuition
of the corresponding semantic space, (ii) demonstrating the robustness,
uniqueness, monotonicity, and isolation of semantic vectors, (iii) providing
an exhaustive empirical evaluation of Sega’s semantic control, and (iv)
showcasing the benefits of Sega over related methods.

##  2 Background

Semantic Dimensions. Research on expressive semantic vectors that allow for
meaningful interpolation and arithmetic pre-date generative diffusion models.
Addition and subtraction on text embeddings such as word2vec
[nikolov2012efficient, nikolov2013Distributed] have been shown to reflect
semantic and linguistic relationships in natural language
[nikolov2012linguistic, vylomova2016take]. One of the most prominent examples
is that the vector representation of ‘King - male + female’ is very close to
‘Queen’. Sega enables similar arithmetic for image generation with diffusion
(cf. Fig. 1(b)). StyleGANs [karras2019style, karras2020analyzing] also
contains inherent semantic dimensions that can be utilized during generation.
For example, Patashnik et al. [patashnik2022styleclip] combined these models
with CLIP [radford2021learning] to offer limited textual control over
generated attributes. However, training StyleGANs at scale with subsequent
fine-tuning is notoriously fragile due to the challenging balance between
reconstruction and adversarial loss. Yet, large-scale pre-training is the base
of flexible and capable generative models [petroni2019language].

Image Diffusion. Recently, large-scale, text-guided DMs have enabled a more
versatile approach for image generation [saharia2022photorealistic,
ramesh2022hierarchical, balaji2022ediffi]. Especially latent diffusion models
[rombach2022High] have been gaining much attention. These models perform the
diffusion process on a compressed space perceptually equivalent to the image
space. For one, this approach reduces computational requirements.
Additionally, the latent representations can be utilized for other downstream
applications [frans2021clipdraw, liu2015deep].

Image Editing. While these models produce astonishing, high-quality images,
fine-grained control over this process remains challenging. Minor changes to
the text prompt often lead to entirely different images. One approach to
tackle this issue is inpainting, where the user provides additionally semantic
masks to restrict changes to certain areas of the image [avrahmi2022blended,
nichol2022glide]. Other methods involve computationally expensive fine-tuning
of the model to condition it on the source image before applying edits
[kawar2022Imagic, valevski2022UniTune]. In contrast, Sega performs edits on
the relevant image regions through text descriptions alone and requires no
tuning.

(a) A (latent) diffusion process inherently organizes concepts and learns
implicitly relationships between them, although there is no supervision.

(b) Guidance arithmetic: Guiding the image ‘a portrait of a king’ (left) using
‘king’$-$‘male’$+$‘female’ results in an image of a ‘queen’ (right).

Figure 1: Semantic guidance (SEGA) applied to the image ‘a portrait of a king’
(Best viewed in color)

Semantic Control. Other works have explored more semantically grounded
approaches for interacting with image generation. Prompt-to-Prompt utilizes
the semantics of the model’s cross-attention layers that attribute pixels to
tokens from the text prompt [hertz2022prompt]. Dedicated operations on the
cross-attention maps enable various changes to the generated image. On the
other hand, Sega does not require token-based conditioning and allows for
combinations of multiple semantic changes. Wu et al. [wu2022uncovering]
studied the disentanglement of concepts for DMs using linear combinations of
text embeddings. However, for each text prompt and target concept, a dedicated
combination must be inferred through optimization. Moreover, the approach only
works for more substantial changes to an image and fails for small edits.
Sega, in contrast, is capable of performing such edits without optimization.

Noise-Estimate Manipulation. Our work is closely related to previous research
working directly on the noise estimates of DMs. Liu et al.
[liu2022Compositional] combine multiple estimates to facilitate changes in
image composition. However, more subtle semantic changes to an image remain
unfeasible with this method. In fact, Kwon et al. [kwon2022diffusion] argue
that the noise-estimate space of DMs is unsuited for semantic manipulation of
the image. Instead, they use a learned mapping function on changes to the
bottleneck of the underlying U-Net. This approach enables various
manipulations that preserve the original image quality. However, it does not
allow for arbitrary spontaneous edits of the image, as each editing concept
requires minutes of training. Sega, in comparison, requires no extension to
the architecture and produces semantic vectors ad-hoc for any textual prompt.
Lastly, Safe Latent Diffusion (SLD) uses targeted manipulation of the noise
estimate to suppress the generation of inappropriate content
[schramowski2022safe]. Instead of arbitrary changes to an image, SLD prevents
one dedicated concept from being generated. Additionally, SLD is complex, and
the hyperparameter formulation can be improved through a deeper understanding
of the numerical properties of DMs’ noise estimate space.

##  3 Semantic Guidance

Let us now devise Semantic Guidance for diffusion models.

###  3.1 Guided Diffusion

The first step towards Sega is guided diffusion. Specifically, diffusion
models (DM) iteratively denoise a Gaussian distributed variable to produce
samples of a learned data distribution. For text-to-image generation, the
model is conditioned on a text prompt $p$ and guided toward an image faithful
to that prompt. The training objective of a DM $\hat{x}_{\theta}$ can be
written as

$\mathbb{E}_{\mathbf{x,c}_{p}\mathbf{,\epsilon},t}\left[w_{t}||\mathbf{\hat{x}}_{\theta}(\alpha_{t}\mathbf{x}+\omega_{t}\mathbf{\epsilon},\mathbf{c}_{p})-\mathbf{x}||^{2}_{2}\right]$
(1)

where $(\mathbf{x,c}_{p})$ is conditioned on text prompt $p$, $t$ is drawn
from a uniform distribution $t\sim\mathcal{U}([0,1])$, $\epsilon$ sampled from
a Gaussian $\mathbf{\epsilon}\sim\mathcal{N}(0,\mathbf{I})$, and
$w_{t},\omega_{t},\alpha_{t}$ influence the image fidelity depending on $t$.
Consequently, the DM is trained to denoise
$\mathbf{z}_{t}:=\mathbf{x}+\mathbf{\epsilon}$ yielding $\mathbf{x}$ with the
squared error loss. At inference, the DM is sampled using the prediction of
$\mathbf{x}=(\mathbf{z}_{t}-\mathbf{\tilde{\epsilon_{\theta}}})$, with
${\tilde{\epsilon_{\theta}}}$ as described below.

Classifier-free guidance [ho2022classifier] is a conditioning method using a
purely generative diffusion model, eliminating the need for an additional pre-
trained classifier. During training, the text conditioning $\mathbf{c}_{p}$
drops randomly with a fixed probability, resulting in a joint model for
unconditional and conditional objectives. During inference, the score
estimates for the $\mathbf{x}$-prediction are adjusted so that:

$\mathbf{\tilde{\epsilon}}_{\theta}(\mathbf{z}_{t},\mathbf{c}_{p}):=\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t})+s_{g}(\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t},\mathbf{c}_{p})-\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t}))$
(2)

with guidance scale $s_{g}$ and $\epsilon_{\theta}$ defining the noise
estimate with parameters $\theta$. Intuitively, the unconditioned
$\epsilon$-prediction is pushed in the direction of the conditioned one, with
$s_{g}$ determining the extent of the adjustment.

###  3.2 Semantic Guidance on Concepts

We introduce Sega to influence the diffusion process along several directions.
To this end, we substantially extend the principles introduced in classifier-
free guidance by solely interacting with the concepts already present in the
model’s latent space. Therefore, Sega requires no additional training, no
extensions to the architecture, and no external guidance. Instead, it is
calculated during the existing diffusion iteration. More specifically, Sega
uses multiple textual descriptions $e_{i}$, representing the given target
concepts of the generated image, in addition to the text prompt $p$.

Intuition. The overall idea of Sega is best explained using a 2D abstraction
of the high dimensional $\epsilon$-space, as shown in Fig. 1. Intuitively, we
can understand the space as a composition of arbitrary sub-spaces representing
semantic concepts. Let us consider the example of generating an image of a
king. The unconditioned noise estimate (black dot) starts at some random point
in the $\epsilon$-space without semantic grounding. The guidance corresponding
to the prompt “a portrait of a king” represents a vector (blue vector) moving
us into a portion of $\epsilon$-space where the concepts ‘male’ and royal
overlap, resulting in an image of a king. We can now further manipulate the
generation process using Sega. From the unconditioned starting point, we get
the directions of ‘male’ and ‘female’ (orange/green lines) using estimates
conditioned on the respective prompts. If we subtract this inferred ‘male’
direction from our prompt guidance and add the ‘female’ one, we now reach a
point in the $\epsilon$-space at the intersection of the ‘royal’ and ‘female’
sub-spaces, i.e., a queen. This vector represents the final direction (red
vector) resulting from semantic guidance.

Isolating Semantics in Diffusion. Next, we investigate the actual noise-
estimate space of DMs on the example of Stable Diffusion (SD). This enables
extracting semantic concepts from within that space and applying them during
image generation.

Numerical values of $\epsilon$-estimates are generally Gaussian distributed.
While the value in each dimension of the latent vector can differ
significantly between seeds, text prompts, and diffusion steps, the overall
distribution always remains similar to a Gaussian distribution (cf. App.
LABEL:app:distribution). Using the arithmetic principles of classifier-free
guidance, we can now identify those dimensions of a latent vector encoding an
arbitrary semantic concept. To that end, we calculate the noise estimate
$\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t},\mathbf{c}_{e})$, which is
conditioned on a concept description $e$. We then take the difference between
$\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t},\mathbf{c}_{e})$ and the
unconditioned estimate $\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t})$ and scale
it. Again, the numerical values of the resulting latent vector are Gaussian
distributed, as shown in Fig. 2. We will demonstrate that those latent
dimensions falling into the upper and lower tail of the distribution alone
encode the target concept. We empirically determined that using only 1-5% of
the $\epsilon$-estimate’s dimensions is sufficient to apply the desired
changes to an image. Consequently, the resulting concept vectors are largely
isolated; thus, multiple ones can be applied simultaneously without
interference (cf. Sec. 4).

Figure 2: Numerical intuition of semantic guidance. The difference between the
concept-conditioned and unconditioned estimates is first scaled. Subsequently,
the tail values represent the dimensions of the specified concept.
Distribution plots calculated using kernel-density estimates with Gaussian
smoothing.

We subsequently refer to the space of these sparse noise-estimate vectors as
semantic space.

One Direction. Let us formally define the intuition for Sega by starting with
a single direction, i.e., editing prompt. Again, we use three
$\epsilon$-predictions to move the unconditioned score estimate
$\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t})$ towards the prompt conditioned
estimate $\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t},\mathbf{c}_{p})$ and
simultaneously away/towards the concept conditioned estimate
$\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t},\mathbf{c}_{e})$, depending on the
editing direction. Formally, we compute
$\mathbf{\bar{\epsilon}}_{\theta}(\mathbf{z}_{t},\mathbf{c}_{p},\mathbf{c}_{e})=$

$\displaystyle\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t})+s_{g}\big{(}\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t},\mathbf{c}_{p})-\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t})\big{)}+\gamma(\mathbf{z}_{t},\mathbf{c}_{e})$
(3)

with the semantic guidance term $\gamma$

$\gamma(\mathbf{z}_{t},\mathbf{c}_{e})=\mu(\psi;s_{e},\lambda)\psi(\mathbf{z}_{t},\mathbf{c}_{e})$
(4)

where $\mu$ applies an edit guidance scale $s_{e}$ element-wise, and $\psi$
depends on the edit direction:

$\psi(\mathbf{z}_{t},\mathbf{c}_{e})=\begin{cases}\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t},\mathbf{c}_{e})-\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t})\quad\text{if
pos. guidance}\\\
-\big{(}\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t},\mathbf{c}_{e})-\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t})\big{)}\quad\text{if
neg. guidance}\end{cases}$ (5)

Thus, changing the guidance direction is reflected by the direction between
$\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t},\\!\mathbf{c}_{e})$ and
$\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t})$.

The term $\mu$ (Eq. 4) considers those dimensions of the prompt conditioned
estimate relevant to the defined editing prompt $e$. To this end, $\mu$ takes
the largest absolute values of the difference between the unconditioned and
concept-conditioned estimates. This corresponds to the upper and lower tail of
the numerical distribution as defined by percentile threshold $\lambda$. All
values in the tails are scaled by an edit scaling factor $s_{e}$, with
everything else being set to 0, such that

$\mu(\psi;s_{e},\lambda)=\begin{cases}s_{e}&\text{where
}|\psi|\geq\eta_{\lambda}(|\psi|)\\\ 0&\text{otherwise}\end{cases}$ (6)

where $\eta_{\lambda}(\psi)$ is the $\lambda$-th percentile of $\psi$.
Consequently, a larger $s_{e}$ increases Sega’s effect.

Sega can also be theoretically motivated in the mathematical background of DMs
[ho2020denoising, Song2021score, kingma2021variational]. The isolated semantic
guidance equation for the positive direction without the classifier-free
guidance term can be written as

$\mathbf{\bar{\epsilon}}_{\theta}(\mathbf{z}_{t},\mathbf{c}_{e})\thickapprox\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t})+\mu(\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t},\mathbf{c}_{e})-\mathbf{\epsilon}_{\theta}(\mathbf{z}_{t}))$
(7)

given Eqs. 3, 4, 5. Further, let us assume an implicit classifier for
$p(\mathbf{c}_{e}|\mathbf{z}_{t})\propto\frac{p(\mathbf{z}_{t},|\mathbf{c}_{e})}{p(z_{t})}$,
where $p(z)$ is the marginal of $z$ for the variance-preserving Markov process
$q(z|x)$ and $x\thicksim p(x)$ [dickstein2015deep]. Assuming exact estimates
$\mathbf{\epsilon}^{*}(\mathbf{z}_{t},\mathbf{c}_{e})$ of
$p(\mathbf{z}_{t}|\mathbf{c}_{e})$ and $\mathbf{\epsilon}^{*}(\mathbf{z}_{t})$
of $p(z_{t})$ the gradient of the resulting classifier can be written as
$\nabla_{\mathbf{z}_{t}}\log
p(\mathbf{c}_{e}|\mathbf{z}_{t})=-\frac{1}{\omega_{t}}\big{(}\mathbf{\epsilon}^{*}(\mathbf{z}_{t},\mathbf{c}_{e})-(\mathbf{\epsilon}^{*}(\mathbf{c}_{e})\big{)}$.
Using this implicit classifier for classifier guidance [dhariwal2021Diffusion]
results in noise estimate
$\mathbf{\bar{{\epsilon^{*}}}}(\mathbf{z}_{t},\mathbf{c}_{e})=\mathbf{\epsilon}^{*}(\mathbf{z}_{t})+w(\mathbf{\epsilon}^{*}(\mathbf{z}_{t},\mathbf{c}_{e})-\mathbf{\epsilon}^{*}(\mathbf{z}_{t}))$.
This is fundamentally similar to Sega as shown in Eq. 7 with $\mu$ isolating
the dimensions of the classifier signal with the largest absolute value (cf.
Eq. 6). However, it should be noted that
$\mathbf{\bar{{\epsilon^{*}}}}(\mathbf{z}_{t},\mathbf{c}_{e})$ is profoundly
different from
$\mathbf{\bar{\epsilon}}_{\theta}(\mathbf{z}_{t},\mathbf{c}_{e})$ as the
latter’s expressions are outputs of an unconstrained network and not the
gradient of a classifier. Consequently, there are no guarantees
[ho2022classifier] for the performance of Sega. Nevertheless, this derivation
provides a solid theoretical foundation and we are able to demonstrate the
effectiveness of Sega empirically in Sec. 5.

To offer even more control over the diffusion process, we make two adjustments
to the methodology presented above. We add warm-up parameter $\delta$ that
will apply guidance $\gamma$ after an initial warm-up period in the diffusion
process, i.e., $\gamma(\mathbf{z}_{t},\mathbf{c}_{e})\\!:=\\!\mathbf{0}\text{
if }t\\!<\\!\delta$. Naturally, higher values for $\delta$ lead to less
substantial adjustments of the generated image. If we aim to keep the overall
image composition unchanged, selecting a sufficiently high $\delta$ ensures
only altering fine-grained output details.

Furthermore, we add a momentum term $\nu_{t}$ to the semantic guidance
$\gamma$ to accelerate guidance over time steps for dimensions that are
continuously guided in the same direction. Hence, $\gamma_{t}$ is defined as:

$\displaystyle\gamma(\mathbf{z}_{t},\mathbf{c}_{e})=\mu(\psi;s_{e},\lambda)\psi(\mathbf{z}_{t},\mathbf{c}_{e})+s_{m}\nu_{t}$
(8)

with momentum scale $s_{m}\in[0,1]$ and $\nu$ being updated as

$\nu_{t+1}=\beta_{m}\nu_{t}+(1-\beta_{m})\gamma_{t}$ (9)

where $\nu_{0}\\!=\\!\mathbf{0}$ and $\beta_{m}\\!\in\\![0,1)$. Thus, larger
$\beta_{m}$ lead to less volatile changes in momentum. Momentum is already
built up during the warm-up period, even though $\gamma_{t}$ is not applied
during these steps.

Beyond One Direction. Now, we are ready to move beyond using just one
direction towards multiple concepts $e_{i}$ and, in turn, combining multiple
calculations of $\gamma_{t}$.

For all $e_{i}$, we calculate $\gamma_{t}^{i}$ as described above with each
defining their own hyperparameter values $\lambda^{i}$, $s_{e}^{i}$. The
weighted sum of all $\gamma_{t}^{i}$ results in

$\hat{\gamma}_{t}(\mathbf{z}_{t},\mathbf{c}_{e_{i}})=\sum\nolimits_{i\in
I}g_{i}\gamma_{t}^{i}(\mathbf{z}_{t},\mathbf{c}_{e_{i}})$ (10)

In order to account for different warm-up periods, $g_{i}$ is defined as
$g_{i}\\!=\\!0$ if $t<\delta_{i}$. However, momentum is built up using all
editing prompts and applied once all warm-up periods are completed, i.e.,
$\forall\delta_{i}:\delta_{i}\geq t$. We provide a pseudo-code implementation
of Sega in App. LABEL:app:implementation and more detailed intuition of each
hyper-parameter along with visual ablations in App. LABEL:app:hyp_ablation.

Sega’s underlying methodology is architecture-agnostic and applicable to any
model employing classifier-free guidance. Consequently, we implemented Sega
for various generative models of different architectures and make our code
available online. If not stated otherwise the examples presented in the main
body, are generated using our implementation based on SD
v1.5222https://huggingface.co/runwayml/stable-diffusion-v1-5 with images from
other models and architectures included in App. LABEL:app:models. We note that
Sega can easily be applied to real images using reconstruction techniques for
diffusion models, which we regard as future work.

(a) Robustness of guidance vectors. Results for guiding towards ‘glasses’ in
various domains without specifying how the concept should be incorporated.

(b) Uniqueness of guidance vectors. The vector for ‘glasses’ is calculated
once on the blue-marked image and subsequently applied to other prompts
(without colored border).

(c) Monotonicity of guidance vectors. The guidance scale for ‘smile’ is
semantically reflected in the images.

Figure 3: Robustness, uniqueness and monotonicity of Sega guidance vectors. In
a) and b) the top row depicts the unchanged image, bottom row depicts the ones
guided towards ‘glasses’. (Best viewed in color)

##  4 Properties of Semantic Space

With the fundamentals of semantic guidance established, we next investigate
the properties of Sega’s semantic space. In addition to the following
discussion, we present further examples in the Appendix.

Robustness. Sega behaves robustly for incorporating arbitrary concepts into
the original image. In Fig. 3(a), we applied guidance for the concept
‘glasses’ to images from different domains. Notably, this prompt does not
provide any context on how to incorporate the glasses into the given image and
thus leaves room for interpretation. The depicted examples showcase how Sega
extracts best-effort integration of the target concept into the original image
that is semantically grounded. This makes Sega’s use easy and provides the
same exploratory nature as the initial image generation.

Uniqueness. Guidance vectors $\gamma$ of one concept are unique and can thus
be calculated once and subsequently applied to other images. Fig. 3(b) shows
an example for which we computed the semantic guidance for ‘glasses’ on the
left-most image and simply added the vector in the diffusion process of other
prompts. All faces are generated wearing glasses without a respective
$\epsilon$-estimate required. This even covers significant domain shifts, as
seen in the one switching from photo-realism to drawings.

However, the transfer is limited to the same initial seed, as
$\epsilon$-estimates change significantly with diverging initial noise
latents. Furthermore, more extensive changes to the image composition, such as
the one from human faces to animals or inanimate objects, require a separate
calculation of the guidance vector. Nonetheless, Sega introduces no visible
artifacts to the resulting images.

Monotonicity. The magnitude of a semantic concept in an image scales
monotonically with the strength of the semantic guidance vector. In Fig. 3(c),
we can observe the effect of increasing the strength of semantic guidance
$s_{e}$. Both for positive and negative guidance, the change in scale
correlates with the strength of the smile or frown. Consequently, any changes
to a generated image can be steered intuitively using only the semantic
guidance scale $s_{e}$ and warm-up period $\delta$. This level of control over
the generation process is also applicable to multiple concepts with arbitrary
combinations of the desired strength of the edit per concept.

Isolation. Different concepts are largely isolated because each concept vector
requires only a fraction of the total noise estimate. Meaning that different
vectors do not interfere with each other. Thus, multiple concepts can be
applied to the same image simultaneously, as shown in Fig. 4. We can see, for
example, that the glasses which were added first remain unchanged with
subsequently added edits. We can utilize this behavior to perform more complex
changes, best expressed using multiple concepts. One example is the change of
gender by simultaneously removing the ‘male’ concept and adding the ‘female’
one (cf. Figs. 5 and LABEL:fig:qualitative_examples).

##  5 Experimental Evaluation

Next, we present exhaustive evaluation of semantic guidance on empirical
benchmarks, as well as additional qualitative tasks. Furthermore, we report
user preferences on direct comparisons with existing methods
[liu2022Compositional, hertz2022prompt, wu2022uncovering]. Our experiments
refute claims of previous research, by demonstrating the suitability of noise
estimates for semantic control [kwon2022diffusion] and its capability for
small, subtle changes [wu2022uncovering]. The comparison to other methods
diffusion-based approaches [hertz2022prompt, wu2022uncovering,
liu2022Compositional] indicate Sega’s improved robustness and capability to
cover a wider range of use cases.

Figure 4: Successive combination of concepts. From left to right an new
concept is added each image. Concepts do not interfere with each other and
only change the relevant portion of the image. (Best viewed in color) Figure
5: Examples from our empirical evaluation benchmark, showing the 10 attributes
edited with Sega. Original and edited images are evaluated by human users one
feature at a time. (Best viewed in color)

Empirical Results. First, we performed an extensive empirical evaluation on
human faces and respective attributes. This setting is inspired by the CelebA
dataset [liu2015deep] and marks a well-established benchmark for semantic
changes in image generation. We generated 250 images with unique seeds using
the prompt ‘an image of the face of a random person’ and manipulated ten
facial attributes. These attributes are a subset of the CelebA labels. All
attributes and respective examples of the corresponding manipulation using
Sega are depicted in Fig. 5. In addition to additive image edits, we evaluated
negative guidance of these attributes as well as two combinations of four
simultaneous edits, as shown in Fig. 4. Therefore, our empirical evaluation
spans 21 attribute changes and combinations in total with all images being
generated using the Stable Diffusion implementation.

We evaluated the generated images with a user study and provide more details
on its implementation in App. LABEL:app:user_study. The results are shown in
Tab. 2. For positive guidance, Sega faithfully adds the target concept to the
image on average in 95% of the cases, with the majority of attributes
exceeding 97%. We further manually investigated the two outliers, ‘Bald’ and
‘Bangs’. We assume that many of the non-native English-speaking annotators
were not familiar with the term ‘bangs’ itself. This assumption is based on
correspondence with some of the workers, and the conspicuously low rate of
annotator consensus. Consequently, the numbers for ‘bangs’ should be taken
with a grain of salt. For baldness, we found that long hair often makes up a
large portion of a portrait and thus requires more substantial changes to the
image. Consequently, such edits require stronger hyperparameters than those
chosen for this study. Furthermore, we observe negative guidance to remove
existing attributes from an image to work similarly well. It is worth pointing
out that the guidance away from ‘beard’, ‘bald’ or ‘gray hair’ usually
resulted in a substantial reduction of the respective feature, but failed to
remove it entirely for $\sim$10% of the images. Again, this suggests that the
hyperparameters were probably not strong enough.

Table 1: Empirical results of our user study conducted on face attributes. Sample sizes result from the portion of the 250 original images that did not contain the target attribute. Annotator consensus refers to the percentage of images for which the majority of annotators agreed on a label. Success rate is reported on those images with annotator consensus. | Attribute | # | Consensus(%) | Success(%)
---|---|---|---|---
Pos. Guidance | Gender |  1241 |  199.2 | 100.0
Glasses |  1243 |  199.6 | 100.0
Smile |  1146 | 100.0 |  199.3
Bald |  1220 |  191.2 |  182.1
Beard |  1135 |  197.8 |  197.0
Hat |  1210 |  199.0 |  199.0
Curls |  1173 |  195.4 |  197.0
Makeup |  1197 |  199.5 |  199.0
Gray hair |  1165 |  197.6 |  191.2
Bangs |  1192 |  186.2 |  182.7
Overall | 1922 |  196.5 |  195.0
Neg. Guidance | No Glasses |  1116 |  198.9 | 100.0
No Smile |  1193 | 100.0 |  194.4
No Bald |  1118 |  194.8 |  183.3
No Beard |  1111 | 100.0 |  189.9
No Hat |  1131 | 100.0 |  190.3
No Curls |  1150 |  198.0 |  198.0
No Makeup |  1121 | 100.0 |  195.2
No Gray Hair |  1152 |  194.6 |  182.7
No Bangs |  1138 |  195.0 |  198.5
Overall |  1420 |  191.9 |  192.1

Table 2: Results of user study on simultaneous combination of face attributes. Sample sizes result from the portion of the 250 original images that did not contain any target attribute. Annotator consensus refers to the percentage of images for which the annotator’s majority agreed on a label. Success rates for combinations on any combination of $x$ attributes with per-attribute scores are reflecting the isolated success of that edit. Nonetheless, all scores are shown for images with all 4 edit concepts applied simultaneously.  | Attribute | # | Consensus(%) | Success(%)
---|---|---|---|---
Combination 1 |  $\mathbf{\geq}$ 1 Attr. |  55 |  198.2 | 100.0
$\mathbf{\geq}$ 2 Attr. | 100.0
$\mathbf{\geq}$ 3 Attr. |  198.2
All 4 Attr. |  190.7
Glasses |  55 |  196.4 | 100.0
Smile | 100.0 |  196.4
Curls | 100.0 |  198.2
Beard | 100.0 | 100.0
Combination 2 |  $\mathbf{\geq}$ 1 Attr. |  45 |  181.8 | 100.0
$\mathbf{\geq}$ 2 Attr. | 100.0
$\mathbf{\geq}$ 3 Attr. | 100.0
All 4 Attr. |  175.6
No Smile |  45 | 100.0 |  197.8
Makeup | 100.0 |  197.6
Hat | 100.0 |  188.6
Female | 100.0 | 100.0

Lastly, we looked into the simultaneous guidance of multiple concepts at once.
The results in Tab. 2 empirically demonstrate the isolation of semantic
guidance vectors. The per-attribute success rate remains similar for four
instead of one distinct edit concept, suggesting no interference of guidance
vectors. Consequently, the success of multiple edits only depends on the joint
probability of the individual concepts. In comparison, if only two out of four
applied concepts were interfering with each other to be mutually exclusive,
the success rate of such a combination would always be 0%. Contrary to that,
we successfully apply concurrent concepts in up to 91% of generated images.

Additionally, we investigated any potential influence of manipulations with
Sega on overall image quality. Utilizing the facial images from the previous
experiment, we calculated FID scores against a reference dataset of FFHQ
[karras2019style]. FID scores for the original, unedited images are remarkable
bad at 117.73, which can be attributed to small artifacts often present in
facial images generated by Stable Diffusion. Editing images with SEGA
significantly improved their quality resulting in an FID score on FFHQ of
59.86. Upon further investigation, we observed that the additional guidance
signal for a dedicated portion of the faces frequently removed uncanny
artifacts, resulting in overall better quality.

Further, we empirically evaluated Sega on the significantly different task of
counteracting inappropriate degeneration [bianchi2022large, birhane2021large,
schramowski2022safe]. For latent DMs, Schramowski et al. [schramowski2022safe]
demonstrated that an additional guidance term may be used to mitigate the
generation of inappropriate images. We demonstrate that Sega’s flexibility
enables it to now perform this task on any diffusion architecture, both latent
and pixel-based. To that end, we evaluated Stable Diffusion v1.5 and
DeepFloyd-IF on the inappropriate-image-prompts (I2P) benchmark
[schramowski2022safe]. Subsequently, we suppressed inappropriate content using
Sega to guide the generation away from the inappropriate concepts as specified
by Safe Latent Diffusion (SLD) [schramowski2022safe]. We empirically chose the
Sega parameters to produce results similar to the ‘Strong’ configuration of
SLD. The results in Tab. 5 demonstrate that Sega performs strong mitigation at
inference for both architectures further highlighting the capabilities and
versatility of the approach.

Comparisons. In addition to the isolated evaluation of Sega’s capabilities, we
conducted randomized user studies directly comparing Sega with related
methods. This comparison investigates the performance of Composable Diffusion
[liu2022Compositional], Promp2Prompt [hertz2022prompt], Disentanglement
[wu2022uncovering], and Sega on tasks from four types of manipulation
categories, reflecting a broad range of operations. Specifically, we
considered 1) composition of multiple edits, 2) minor changes, 3) style
transfer and 4) removal of specific objects from a scene. For each method and
task, we selected the best-performing hyperparameters on a set of seeds < 100
and subsequently generated non-cherry-picked edits on a fixed test set (i.e.,
first applicable seeds >= 100). As first evaluation step, we conducted a user
study to assess the success of an edit based on the presence/absence of the
target attribute(s) as shown in Tab. LABEL:tab:comp_study. The comparatively
low scores on removal tasks result from the methods often reducing the
presence of the targeted objects but not eliminating all of them in some
cases. However, this peculiarity of the evaluation affects all methods
similarly and does not influence the comparability of capabilities. The
results demonstrate that Sega clearly outperforms Prompt2Prompt and
Disentanglement on all examined editing tasks. Compared to Composable
Diffusion, SEGA again has significantly higher success rates for multi-
conditioning and minor changes while achieving comparable performance for
style transfer and object removal.

Beyond the simple success rate of edits, we evaluated the faithfulness to the
original image composition. To that end, users considered pair-wise
comparisons of samples that both methods edited successfully and assessed the
similarity with the original image. Importantly, users strongly prefer SEGA
results over Composable Diffusion for 83.33% (vs. 13.33%) of samples. These
two studies highlight that Sega is generally preferred in terms of its edit
capabilities and perceived fidelity over other methods. We present examples
and further details on comparisons to related diffusion and StyleGAN
techniques in App. LABEL:app:comps.

Qualitative Results. In addition to the empirical evaluation, we present
qualitative examples on other domains and tasks. We show further examples in
higher resolution in the Appendix. Overall, this highlights the versatility of
Sega since it allows interaction with any of the abundant number of concepts
diffusion models are capable of generating in the first place. We performed a
diverse set of style transfers, as shown in Fig. LABEL:fig:style_transfer_v2.
Sega faithfully applies the styles of famous artists, as well as artistic
epochs and drawing techniques. In this case, the entirety of the image has to
be changed while keeping the image composition the same. Consequently, we
observed that alterations to the entire output—as in style transfer—require a
slightly lower threshold of $\lambda\approx 0.9$. Nonetheless, this still
means that 10% of the $\epsilon$-space is sufficient to change the entire
style of an image. Fig. LABEL:fig:style_transfer_v2 also includes a comparison
between outputs produced by Sega with those from simple extensions to the
prompt text. Changing the prompt also significantly alters the image
composition. These results further highlight the advantages of semantic
control, which allows versatile and yet robust changes.

Table 3: Results of Stable Diffusion and IF on the I2P benchmark [schramowski2022safe]. Values reflect the probability of generating inappropriate content (the lower the better) with respect to the joint Q16/NudeNet classifier proposed by Safe Latent Diffusion[schramowski2022safe]. Subscript values denote the expected maximum inappropriateness over 25 prompts (the lower the better). For both architectures Sega performs strong mitigation at inference. | Harassment | Hate |  Illegal Activity | Self-Harm | Sexual | Shocking | Violence | Overall
---|---|---|---|---|---|---|---|---
SD |  |  |  |  |  |  |  |


# Search for new physics in $b\to q\ell\ell$ decays

F. Dettori 111On behalf of the LHCb collaboration Oliver Lodge Laboratory,
University of Liverpool, Liverpool, UK

###### Abstract

Recent results obtained in experiments at the LHC in the field of rare
$b$-hadron decays are reviewed in this contribution, with a focus on $b\to
q\ell\ell$ processes. A general status is presented as well as recently
completed measurements.

##  1 Introduction

The study of flavour-changing neutral-current processes (FCNC) is a sensitive
tool to explore possible interactions beyond those predicted by the Standard
Model (SM). In particular, rare hadron decays having tiny probabilities in the
SM, can receive possible New Physics (NP) contributions at the same level or
even larger than the SM. Probing NP using these techniques is especially
sensitive when the SM predictions are very precise; in these cases, we can
often probe energy scales orders of magnitude higher than those available for
the direct production of new states at colliders. In addition, rare decays
offer the possibility of model-independent tests of physics beyond the SM.
Finally, historically rare decays have been the laboratory of many particle
physics discoveries.

The description of heavy-quark hadrons rare decays is usually made in terms of
effective field theories where SM and NP contributions are computed in terms
of local operators and their Wilson coefficients. This allows to put model-
independent constraints on possible NP couplings. As an example possible new
(pseudo)-scalar contributions in $b\to q\ell^{+}\ell^{-}$ processes can be
defined as a term in the effective hamiltonian proportional to
$C_{S(P)}\bar{q}_{L}b_{L}\bar{\ell}(\gamma_{5})\ell$ where $C_{S(P)}$ are the
Wilson coefficients, and the overall normalisation is not reported. Similarly
in what follows it is useful to define (axial-)vector contributions as
$C_{9(10)}\bar{q}_{L}\gamma^{\mu}b_{L}\bar{\ell}\gamma_{\mu}(\gamma_{5})\ell$.
We do not list them, but additional operators also possibly contribute to
$b\to q\ell^{+}\ell^{-}$ transitions as well as the chirally-flipped versions
of the ones defined.

In this contribution we review recent results in rare $b$-hadron decays based
on quark-level $b\to s\ell^{+}\ell^{-}$ and $b\to d\ell^{+}\ell^{-}$
transitions with a focus on measurements from experiments at the Large Hadron
Collider (LHC).

##  2 Scalar, pseudoscalar and axial-vector couplings

NP contributions in the form of scalar, pseudoscalar and axial-vector
couplings are best tested with $B^{0}_{(s)}\to\ell^{+}\ell^{-}$ decays. The
branching fractions of these decays are very well predicted in the SM [1, 2]
and thus allow for extremely sensitive tests of NP. Due to helicity
suppression the $B^{0}_{(s)}\to e^{+}e^{-}$ decays are too rare to be
currently probed at branching fractions close to the SM contribution. The
$B^{0}_{(s)}\to\tau^{+}\tau^{-}$ decays are the less rare but, due to the
difficulty of reconstructing $\tau$ leptons, especially at hadron colliders,
the SM branching fraction is still out of reach. The current world best limits
on these decays have been recently set by LHCb [3]. Nevertheless it is
important to probe these decays, especially in light of possible hints of
lepton non universality (see C. Langenbruch contribution at this same
conference [4]). The $B^{0}_{d,s}\to\mu^{+}\mu^{-}$ decays instead, being
cleaner from an experimental point of view, are being probed by several
experiments. The first evidence for the $B^{0}_{s}\to\mu^{+}\mu^{-}$ decay,
came from the LHCb experiment [5], while the definitive observation resulted
from the combined analysis of the LHCb and CMS Run 1 measurements [6]. The
current world best measurement comes from the last LHCb measurement [7], where
the full Run 1 as well as 1.4 $fb^{-1}$ of integrated luminosity collected in
Run 2 was exploited leading to the first observation of the
$B^{0}_{s}\to\mu^{+}\mu^{-}$ decay from a single experiment. The branching
fraction measurement yields ${\cal B}(B^{0}_{s}\to\mu^{+}\mu^{-})=\left(3.0\pm
0.6^{+0.3}_{-0.2}\right)\times 10^{-9}$, in good agreement with recent SM
predictions [1, 2] as well as with the previous measurements. The Run 1 ATLAS
search for the same decay is instead in slight tension with these results [8],
reporting only a 1.4$\sigma$ excess over background expectations and thus
putting an upper limit on the branching fraction at ${\cal
B}(B^{0}_{s}\to\mu^{+}\mu^{-})<3.0\times 10^{-9}$ at 90% CL, lower than the SM
prediction. For more details on these and other very rare decays measurements
see M. van Veghel contribution at this same conference [9].

##  3 Vector couplings and a possible anomaly

While scalar and pseudo-scalar couplings seem to be in good agreement with the
SM, the same is not true for other couplings, especially vector ones
($C_{9}$). Several small discrepancies, which we review briefly, are
accumulating giving an overall tension with the SM.

We start from the $B^{0}_{d}\to\bar{K}^{\ast 0}\mu^{+}\mu^{-}$ decay; its rich
phenomenology allows to test several couplings simultaneously through the
study of its angular distributions. Recent measurements have been done by four
experiments, often exploiting the so-called optimised observables [10], which
have the advantage of reduced theoretical uncertainty. Within the Run 1 LHCb
measurement [11] the full fit to CP-averaged observables shows a tension with
the SM with a significance of 3.4 standard deviations. Considering single
variables, the most striking feature is a discrepancy in the $P^{\prime}_{5}$
parameter. This behaviour seems to be confirmed by ATLAS [12] and Belle [13]
measurements; while the CMS [14] measurement is more in agreement with SM.
This is shown in Figure 1, where the different measurements are presented as a
function of $q^{2}$, the dimuon invariant mass squared, and compared with SM
predictions [15, 16, 17].

Figure 1: Measurements of the $F_{L}$ and $P^{\prime}_{5}$ observables in
$B^{0}_{d}\to\bar{K}^{\ast 0}\mu^{+}\mu^{-}$ decays, in bins of $q^{2}$, for
different experiments and compared with SM predictions.

Similarly, discrepancies are present in the branching fractions of several
$b\to s\mu^{+}\mu^{-}$ processes, recently measured by LHCb, which tend to
show a mild tension with the SM. In particular the differential and integrated
branching fractions of $B^{+}\to K^{+}\mu^{+}\mu^{-}$, $B^{+}\to
K^{\ast+}\mu^{+}\mu^{-}$, $B^{0}\to K^{0}\mu^{+}\mu^{-}$ [18] as well as
$B^{0}_{d}\to\bar{K}^{\ast 0}\mu^{+}\mu^{-}$ [19] and
$B^{0}_{s}\to\phi\mu^{-}\mu^{+}$ [20] lie all below the SM prediction, albeit
not with great significance.

These discrepancies summed with the hints of lepton flavour universality
violation in $B^{+}\to K^{+}\ell^{+}\ell^{-}$, $B^{0}_{d}\to\bar{K}^{\ast
0}\ell^{+}\ell^{-}$ and $B\to D^{(\ast)}\ell\nu$ decays [4] lead to possible
global discrepancies with respect to the SM even above the 5$\sigma$ level. In
particular, global fits [21, 22, 23, 24] could indicate possible NP
contributions on the $C_{9}$ Wilson coefficient, or to the $C_{9}$ and
$C_{10}$ simultaneously. The significance of these discrepancies is almost
completely dominated by the statistical uncertainties, hence they will need to
be confirmed with additional statistics by the same experiments, and cross-
checked by other experiments (such as Belle-II), but give encouraging hints on
the direction to follow.

###  3.1 Angular analysis of $B^{+}\to K^{+}\mu^{+}\mu^{-}$ at CMS

Presented for the first time at this conference, the CMS collaboration has
performed an angular analysis of the $B^{+}\to K^{+}\mu^{+}\mu^{-}$ decays,
exploiting 20.5$fb^{-1}$ of integrated luminosity collected in Run 1 at
$\sqrt{s}=8$ TeV [25]. The differential decay rate of this decay can be
written as:

$\frac{1}{\Gamma_{\ell}}\frac{d\Gamma_{\ell}}{d\cos\theta_{\ell}}=\frac{3}{4}(1-F_{H})(1-\cos^{2}\theta_{\ell})+\frac{1}{2}F_{H}+A_{FB}\cos\theta_{\ell}\quad,$
(1)

where $\theta_{\ell}$ is the angle between the direction of the $\mu^{-}$ and
the one of the $K^{+}$ meson for the $B^{+}$ decay. The two parameters on
which the differential decay rate depends ($A_{FB}$ and $F_{H}$) are a
function of $q^{2}$ and sensitive probes to scalar and tensor new physics
contributions. The $B^{+}\to K^{+}\mu^{+}\mu^{-}$ angular analysis was
previously performed also by the LHCb [26], BaBar [27], Belle [28] and CDF
[29] experiments. The CMS results as a function of $q^{2}$ are shown in Figure
2, as extracted from about 2300 signal candidates. The largest systematic
uncertainties are due to modelling of the background components and dominate
the total uncertainty in some bins. The results are compatible with previous
measurements and with SM predictions. While this measurement is not yet the
world most precise, it is important to have measurements from different
experiments, especially when anomalies such as the ones described are
surfacing. A measurement of the $B^{+}\to K^{+}\mu^{+}\mu^{-}$ differential
branching fraction by CMS is therefore important and awaited.

Figure 2: Measurement of (left) the $A_{FB}$ parameter and (right) the $F_{H}$
parameter as a function of $q^{2}$ in the angular analysis of $B^{+}\to
K^{+}\mu^{+}\mu^{-}$ decays in CMS.

###  3.2 $b\to d\ell\ell$

Processes involving $b\to d\ell^{+}\ell^{-}$ transitions are even more rare
than those with an $s$ quark due to the additional CKM suppression. The
observation of these processes was established by LHCb in
$B^{+}\to\pi^{+}\mu^{+}\mu^{-}$ and $\Lambda_{b}^{0}\to
p\pi^{-}\mu^{+}\mu^{-}$ decays [30, 31]. The $B^{0}_{s}\to\bar{K}^{\ast
0}\mu^{+}\mu^{-}$ decay is instead yet to be observed. This decay can be
sensitive to NP contributions, but is also an important probe of the
$|V_{td}|/|V_{ts}|$ ratio. A search for this decay has been performed by LHCb
exploiting 4.6$fb^{-1}$ collected at different energies in Run 1 and 2 [32]. A
common selection for the $B^{0}_{s}\to\bar{K}^{\ast 0}\mu^{+}\mu^{-}$ and the
control and normalisation channels $B^{0}_{d}\to\bar{K}^{\ast
0}\mu^{+}\mu^{-}$, $B^{0}_{d}\to\bar{K}^{\ast 0}J/\psi$ and
$B^{0}_{s}\to\bar{K}^{\ast 0}J/\psi$ is devised. The search for the rare non-
resonant channel is restricted to dimuon invariant masses squared of
$0.1<q^{2}<19.0\leavevmode\nobreak\ \rm{GeV}^{2}/c^{4}$ excluding the region
$12.5<q^{2}<15.0\leavevmode\nobreak\ \rm{GeV}^{2}/c^{4}$, which is dominated
by charmonium resonances. After a loose pre-selection, candidates are
classified in bins of a neural network based on geometric and kinematic
variables. The background is predominantly composed of combinatorial
background and of the upper tail of the $B^{0}_{d}\to\bar{K}^{\ast
0}\mu^{+}\mu^{-}$ distribution. The distribution of the
$K^{-}\pi^{+}\mu^{+}\mu^{-}$ candidates summed over the three most sensitive
bins of neural network is shown in Figure 3. The presence of an excess on top
of the background at the $B^{0}_{s}$ mass can be seen. From a simultaneous fit
a combined significance of 3.4 standard deviations is derived, including
systematic uncertainties, which represents the first evidence of the
$B^{0}_{s}\to\bar{K}^{\ast 0}\mu^{+}\mu^{-}$ decay. Normalising the signal to
the $B^{0}_{d}\to\bar{K}^{\ast 0}\mu^{+}\mu^{-}$ decay a branching fraction of
$\mathcal{B}(B^{0}_{s}\to\bar{K}^{\ast 0}\mu^{+}\mu^{-})=(2.9\pm 1.0\pm 0.2\pm
0.3)\times 10^{-8}$ is obtained, where the first uncertainty is statistical,
the second systematic and the third is due to limited knowledge of parameters
used in the normalisation. This result is in agreement with SM expectations.

Figure 3: Invariant mass distribution for (left) $K^{-}\pi^{+}\mu^{+}\mu^{-}$
candidates for the three most sensitive neural network bins of the LHCb
analysis and (right) a zoom on the same plot to highlight the
$B^{0}_{s}\to\bar{K}^{\ast 0}\mu^{+}\mu^{-}$ signal.

###  3.3 $b$-baryons

A different look to $b\to s\ell^{+}\ell^{-}$ transitions can be taken by
studying $b$-baryon decays, the rare decays of which start to be probed only
recently. In addition to the already mentioned $\Lambda_{b}^{0}\to
p\pi^{-}\mu^{+}\mu^{-}$, the LHCb collaboration has studied the
$\Lambda_{b}^{0}\to pK^{-}\mu^{+}\mu^{-}$ decay exploiting 3$fb^{-1}$ of Run 1
data [33]. The $\Lambda_{b}^{0}\to pK^{-}\mu^{+}\mu^{-}$ decay is observed for
the first time, with large significance. Exploiting a signal yield of $600\pm
44$ candidates, a search for CP violation is performed in this channel which
can be a sensitive probe of NP [34]. A measurement of CP violation is built
from raw yields as:

$\mathcal{A}_{\rm{CP}}\propto\mathcal{A}_{\rm{raw}}=\frac{N(\Lambda_{b}^{0}\to
pK^{-}\mu^{+}\mu^{-})-N(\bar{\Lambda}_{b}^{0}\to\bar{p}K^{+}\mu^{+}\mu^{-})}{N(\Lambda_{b}^{0}\to
pK^{-}\mu^{+}\mu^{-})+N(\bar{\Lambda}_{b}^{0}\to\bar{p}K^{+}\mu^{+}\mu^{-})}\quad,$
(2)

where the proper conversion to $\mathcal{A}_{\rm{CP}}$ is obtained correcting
for production and reconstruction asymmetries. For additional robustness, CP
violation is searched in the difference of this observable between the
$\Lambda_{b}^{0}\to pK^{-}\mu^{+}\mu^{-}$ and the control channel
$\Lambda_{b}^{0}\to pK^{-}J/\psi$, on which no significant CP violation is
expected. This measurement results in: $\Delta\mathcal{A}_{C\\!P}=(-3.5\pm
5.0(stat)\pm 0.2(syst))\times 10^{-2}$, showing no sign of CP violation. A
second CP violating variable is constructed from the positive-negative
asymmetry of the triple products
$C_{\hat{T}}=\vec{p}_{\mu^{\pm}}\cdot\vec{p}_{p}\times\vec{p}_{K^{+}}$;
comparing the positive and negative muon CP-odd and P-odd observables are
obtained. The results for the CP-odd observable are
$a_{C\\!P}^{\widehat{T}-odd}=(1.2\pm 5.0(stat)\pm 0.7(syst))\times 10^{-2}$,
showing again no sign of CP violation, in agreement with SM predictions [35,
36].

##  4 Conclusions

A very brief account of recent rare decays searches and measurements, in the
field of $b\to q\ell^{+}\ell^{-}$ transitions was given in this contribution.
The topic is among the most exiting in current particle physics as small
discrepancies that point towards possible NP contributions beyond the SM are
accumulating. The results based on the analysis of Run 1 LHC data by the LHCb,
ATLAS and CMS experiments show tantalising hints of possible deviations from
the SM, especially in the muonic vector coupling. These discrepancies will be
possibly confirmed or disproved by Run 2 data and through different
observables. Finally, the LHC experiments will face soon the healthy
competition of the Belle II experiment in the run for new physics.

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