# Hall-Littlewood Polynomials, Springer Fibers and the Annihilator Varieties
of Induced Representations

Zhuohui Zhang
Weizmann Institute of Science<EMAIL_ADDRESS>
###### Abstract.

For $G=GL(n,\mathbb{C})$ and a parabolic subgroup $P=LN$ with a two-block Levi
subgroup $L=GL(n_{1})\times GL(n_{2})$, the space
$G\cdot(\mathcal{\mathcal{O}}+\mathfrak{n})$, where $\mathcal{O}$ is a
nilpotent orbit of $\mathfrak{l}$, is a union of nilpotent orbits of
$\mathfrak{g}$. In the first part of our main theorem, we use the geometric
Sakate equivalence to prove that $\mathcal{O^{\prime}}\subset
G\cdot(\mathcal{\mathcal{O}}+\mathfrak{n})$ if and only if some Littlewood-
Richardson coefficients do not vanish. The second part of our main theorem
describes the geometry of the space $\mathcal{O}\cap\mathfrak{p}$, which is an
important space to study for the Whittaker supports and annihilator varieties
of representations of $G$.

##  1\. Introduction

A nilpotent orbit of a reductive group $G$ over $\mathbb{C}$ is an orbit of
the adjoint or the coadjoint action of $G$ on the nilpotent cone
$\mathcal{N}\subset\mathfrak{g}$ or $\mathfrak{g}^{*}$, respectively. The set
of nilpotent orbits $\mathcal{O}\subset\mathcal{N}$ forms a poset with a
partial order $\mathcal{O}_{1}<\mathcal{O}_{2}$ defined by the closure order
$\mathcal{O}_{1}\subset\overline{\mathcal{O}_{2}}$. For classical groups,
there is a bijection between the nilpotent orbits and certain integer
partitions. In particular, for $GL(n)$ and $SL(n)$, each partition $\alpha$ of
the integer $n$ corresponds to a nilpotent orbit $\mathcal{O}_{\alpha}$. The
counterpart of the closure order
$\mathcal{O}_{\alpha}\subset\overline{\mathcal{O}_{\beta}}$ on partitions is
given by

$\alpha<\beta\text{ if and only if
}\sum_{i=1}^{k}\alpha_{i}\leq\sum_{i=1}^{k}\beta_{i}\text{ for all }k.$

The notion of _induced orbits_ provides a connection between the nilpotent
orbits of a subgroup and the orbits of the ambient group. Following [CM93,
Chapter 7], for any nilpotent orbit $\mathcal{O}$ of a Levi subgroup $L\subset
G$, there is an unique nilpotent orbit of $G$ which intersects
$\mathcal{O}+\mathfrak{n}$ in a Zariski-open subset. This orbit is called the
_induced orbit_ $\mathrm{ind}_{\mathfrak{l}}^{\mathfrak{g}}(\mathcal{O})$ in
$G$. At the other end, we can define the _Bala-Carter_ inclusion
$\mathrm{inc_{\mathfrak{l}}^{\mathfrak{g}}(\mathcal{O})}$ of a nilpotent orbit
$\mathcal{O}$ of $\mathfrak{l}$ as the $G$-saturation $G\cdot\mathcal{O}$ in
$\mathfrak{g}$. These two orbits are the maximal and minimal elements in the
poset which consists of all the $G$-orbits contained in
$G\cdot(\mathcal{O}+\mathfrak{n})$. The first part of the main result of this
paper describes this poset of the nilpotent orbits contained in
$G\cdot(\mathcal{O}+\mathfrak{n})$.
Another degree of freedom for the set $G\cdot(\mathcal{O}+\mathfrak{n})$ is
the choice of parabolic subgroups $P$ and the nilpotent radicals $N\subset P$.
For each pair of orbits $\mathcal{O}_{\gamma}$ of $\mathfrak{l}$ and
$\mathcal{O}_{\alpha}$ of $\mathfrak{g}$, we will also describe an algebraic
variety which parametrizes all the parabolic subgroups $P=LN$ such that

$\mathcal{O}_{\alpha}\subset
G\cdot\left(\mathcal{O}_{\gamma}+\mathfrak{n}\right).$

The collection of all such conjugate parabolic subgroups forms a closed
subvariety of the partial flag manifold $G/P$, and some information of its
topology is encoded in the Littlewood-Richardson coefficients and the Hall-
Littlewood polynomials. This variety is closely related to the geometry of
$\mathcal{O}_{\gamma}\cap\mathfrak{p}$, and we will provide this result as the
second part of our main theorem.
In Secion 5, we will also discuss the application of this geometric result to
the wavefront set of the subrepresentations of an induced representation,
following the result in [GS20].
Now we state the main theorem of this paper:

######  Theorem A.

For $G=GL(n,\mathbb{C})$ and a parabolic subgroup $P=LN$ with a standard Levi
subgroup $L=GL(n_{1},\mathbb{C})\times GL(n_{2},\mathbb{C})\subset G$ embedded
block-diagonally, let $\alpha$ be a partition of $n_{1}$ and $\beta$ be a
partition of $n_{2}$, consider the nilpotent orbit
$\mathcal{O}_{\alpha,\beta}=\mathcal{O}_{\alpha}\times\mathcal{O}_{\beta}$ of
the subgroup $L$ where $\mathcal{O}_{\alpha},\mathcal{O}_{\beta}$ are the
nilpotent orbits corresponding to the partitions $\alpha,\beta$ in $GL(n_{1})$
and $GL(n_{2})$, repsectively, then:

  1. (1)

A $G$-orbit $\mathcal{O}_{\gamma}$ is contained in
$G\cdot(\mathcal{O}_{\alpha,\beta}+\mathfrak{n})$ if and only if the
Littlewood-Richardson coefficient $c_{\alpha,\beta}^{\gamma}\neq 0$.

  2. (2)

The Littlewood-Richardson coefficient is the number of irreducible components
of the subvariety

$\mathcal{P}_{\alpha,\beta}^{\gamma}=\left\\{x\in\mathcal{O_{\gamma}}\cap\mathfrak{p}\mid
p_{\mathfrak{l}}(x)\in\mathcal{O}_{\alpha}\times\mathcal{O}_{\beta}\right\\}\subset\mathcal{O}_{\gamma}\cap\mathfrak{p}$

where the map $p_{\mathfrak{l}}$ is the projection
$p_{\mathfrak{l}}:\mathfrak{p}\mapsto\mathfrak{p}/\mathfrak{n}\cong\mathfrak{l}$
onto the Levi subgroup.

The first part of this theorem is a direct consequence of the theory of Hall
polynomials which was discussed for abelian $p$-groups in [Mac98, Appendix
Chapter II, Theorem AZ.1], and can be proven as a corollary of the Hall’s
theorem in [Kle69, Theorem 2.2]. The proof of Hall’s theorem is combinatorial
and makes use of Schubert calculus. However, in order to find an interpretion
of the Littlewood-Richardson coefficients in our context, we would like to
give a geometric proof to the first part of the theorem as a corollary of the
geometric Satake isomorphism. The geometric constructions used in the proof of
the first part will shed light on the proof of the second part of Theorem A in
Section 4.
It is worth mentioning that since we can induce the orbits by stages, the
first part of Theorem A can be generalized to arbitrary standard parabolic
subgroups:

######  Corollary B.

For any partition $\underline{n}=(n_{1},\ldots,n_{r})$ of $n$ and partitions
$\alpha_{i}$ of the integers $n_{i}$, we take a standard parabolic subgroup
$P_{\underline{n}}=L_{\underline{n}}N_{\underline{n}}$ with a Levi subgroup
isomorphic to $GL(n_{1})\times\ldots\times GL(n_{r})$. A nilpotent orbit
$\mathcal{O}_{\gamma}$ of $GL(n)$ is contained in
$G\cdot(\prod\mathcal{O}_{\alpha_{i}}+\mathfrak{n}_{\underline{n}})$ if and
only if the number

$N^{\gamma}_{\alpha_{1},\ldots,\alpha_{r}}=\sum_{\beta_{2},\ldots,\beta_{r-1}}c_{\alpha_{1},\alpha_{2}}^{\beta_{2}}c_{\beta_{2},\alpha_{3}}^{\beta_{3}}c_{\beta_{3},\alpha_{4}}^{\beta_{4}}\ldots
c_{\beta_{r-1},\alpha_{r}}^{\gamma}$

is nonzero.

In terms of Schur functions, the number
$N_{\alpha_{1},\ldots,\alpha_{r}}^{\gamma}$ is the coefficient of
$\mathbf{s}_{\gamma}$ in the product of the Schur polynomials
$\mathbf{s}_{n_{1}}\ldots\mathbf{s}_{n_{r}}$.

### Acknowledgements

I thank Dmitry Gourevitch for his support, enlightening discussions and
valuable comments about this project. This research is supported by the Israel
Science Foundation (ISF) 249/17 and ERC Starting Grants 637912.

##  2\. Preliminaries and Notation

In this section, we specify the notations which are used throughout this
paper. Let $G$ be a semisimple algebraic group over $\mathbb{C}$, we denote by

  * •

$T\subset B\subset G$ a choice of maximum torus $T$ and a Borel subgroup $B$,

  * •

$X_{*}(T)$ the cocharacter lattice, and $X^{*}(T)$ the character lattice of
$T$,

  * •

$\Delta_{G}(T)$ the set of roots, $\Delta_{G,+}(T)$ the set of positive roots
and $\Delta_{G,s}(T)$ the set of simple roots, and $\Delta^{G}(T)$ the set of
coroots.

We can identify the character lattice as the integer lattice $\mathbb{Z}^{n}$
where $n$ is the rank of $G$. Each partition
$\lambda=(\lambda_{1},\ldots,\lambda_{n})$ corresponds to the character

$(t_{1},\ldots,t_{n})\mapsto t_{1}^{\lambda_{1}}\ldots t_{n}^{\lambda_{n}}.$

In particular, under this correspondence, the half sum of positive roots

$\rho=\frac{1}{2}\sum_{\alpha\in\Delta_{G,+}}\alpha$

corresponds to the partition $(n-1,n-2,\ldots,1,0)$. Fixing a maximal torus
$T$, the quadruple $(X^{*},\Delta^{G},X_{*},\Delta_{G})$ is called the _root
datum_ of $G$. Its dual root datum $(X_{*},\Delta_{G},X^{*},\Delta^{G})$ is
obtained by switching the character lattice/roots and the cocharacter
lattice/coroots. The algebraic group corresponding to the dual root datum is
called the _Langlands dual group_ and is denoted by $\check{G}$.

###  2.1. Young Diagrams

In this section we will summarize the basic concepts related to Young diagrams
and define the Littlewood-Richardson coefficients. A partition
$\alpha=(\alpha^{1},\ldots,\alpha^{r})$ of an integer $n$ corresponds to a
Young diagram (whose $i$-th row has length $\alpha^{i}$) of the shape
$\alpha$. In a _Young tableau_ of the shape $\alpha$, integers, starting with
1, are filled into the boxes of the Young diagram $\alpha$. We denote the set
of all Young tableaux of the shape $\alpha$ by $\mathcal{T}_{\alpha}$. A Young
tableau is called _standard_ if the entries of each box are strictly
increasing from the left to right in each row, and increasing from top to
bottom in each column. If the entries are weakly increasing along the rows,
the tableau is called a _semistandard_ tableau. It is worth noting that each
standard Young tableau $T$ of the shape $\alpha$ corresponds to a sequence of
Young diagrams

$0=\alpha_{0}\subset\alpha_{1}\subset\ldots\subset\alpha_{r}=\alpha$

with each $\alpha_{i}$ formed by the boxes in $T$ with contents $\leq i$. For
example, the Young tableau

$T=$ smalltableaux 1 & 1 2

1 3

2

corresponds to the following sequence of Young diagrams:

$\emptyset\subset$ 2,1 $\subset$ 3,1,1 $\subset$ 3,2,1.

Similar to the usual Young diagrams discussed above, for each pair of Young
diagrams $(\alpha,\gamma)$ such that $\alpha\subset\gamma$, a _skew Young
diagram_ is the complement of $\alpha$ in $\gamma$, with the two diagrams
aligned along their top and left edges. For example, for $\alpha=(2,1)$ and
$\gamma=(4,3,2)$, the skew Young diagram $\gamma/\alpha$ looks like

2+2,1+2,0+2.

We denote the set of all semistandard skew tableaux of the shape
$\gamma/\alpha$ by $\mathcal{T}_{\gamma/\alpha}$. Similar to the
correspondence between Young tableaux and sequences of Young diagrams
described above, any skew tableau $T\in\mathcal{T}_{\gamma/\alpha}$
corresponds to a sequence of Young diagrams from $\alpha$ to $\gamma$, whose
$i$-th stage $\alpha_{i}$ consists of the boxes in $T$ such that the entries
in each of the selected boxes are $\leq i$. Assuming the maximal content of
the tableaux is $r$, there is a sequence

$\alpha=\alpha_{0}\subset\alpha_{1}\subset\alpha_{2}\subset\ldots\subset\alpha_{r}=\gamma.$

The entries of such a skew tableau also gives us a new partition

$\beta=(|\alpha_{1}/\alpha_{0}|,\ldots,|\alpha_{r}/\alpha_{r-1}|).$

For example, the skew tableau

smalltableaux 11,2,12,12

yields the partition $\beta=(4,3)$. We say a tableau $T$, or a corresponding
sequence of Young diagrams from $\alpha$ to $\gamma$ has type
$(\alpha,\beta;\gamma)$ if this new partition is $\beta$.

###  2.2. The Littlewood-Richardson Rule

For each pairs of Young diagrams $(\alpha,\gamma)$ such that
$\alpha\subset\gamma$, a semistandard skew Young tableau is _Littlewood-
Richardson_ if its _reverse lattice words_ , i.e. the string of content of
each box, read from top to bottom and from right to left like in Hebrew,
satisfies the _Yamanouchi condition_ , i.e. for each content $i$, the number
of $i$’s in the length $k$ prefix of its reverse lattice word is great or
equal to the number of $i+1$’s in the length $k$ prefix. For example, the
reverse lattice word of the tableau

smalltableaux & 11

1 2

2 3

is $112132$. It satisfies the Yamanouchi condition.
For each triple of Young diagrams $(\alpha,\beta;\gamma)$ such that
$\alpha,\beta\subset\gamma$, the Littlewood-Richardson coefficient
$c_{\alpha,\beta}^{\gamma}$ is defined as the number of Littlewood-Richardson
tableaux of type $(\alpha,\beta;\gamma)$.
The usual combinatorial description of the Littlewood-Richardson coefficient
is given by the Hall-Littlewood polynomials $G^{\gamma}_{\alpha,\beta}(q)$.
For a pair $(\mathcal{O},\mathfrak{p})$ consisting of a discrete valuation
ring $\mathcal{O}$ and its maximal ideal $\mathfrak{p}$ with a residue field
of size $q$, and a triple of partitions $(\alpha,\beta;\gamma)$, the Hall-
Littlewood polynomial $G^{\gamma}_{\alpha,\beta}(q)$ is defined as the number
of submodules $N\subset M_{\gamma}$ such that $N\cong M_{\alpha}$ and
$M_{\gamma}/N\cong M_{\beta}$. The Hall polynomial
$G^{\gamma}_{\alpha,\beta}(q)$ has degree
$\langle\rho,\alpha+\beta-\gamma\rangle$, and its top-degree coefficient is
equal to the Littlewood-Richardson coefficient $c^{\gamma}_{\alpha,\beta}$. A
detailed proof of this fact is in the appendix of [Mac98, Chapter 2].
However, it’s important to have a geometric model of the abelian group
extensions so that we can get more information related to nilpotent orbits. In
the following sections, we will introduce the theory of affine Grassmannian
and a geometric interpretation of the Littlewood-Richardson coefficient.

##  3\. Lattices and Affine Grassmannian

In this paper, we will introduce four major pictures for the Littlewood-
Richardson coefficient: the first picture is the extensions of torsion
$\mathbb{C}[t]$-modules and the top-degree coefficient of the Hall-Littlewood
polynomial, which we have already introduced in the previous section. The
second picture is the extension of sublattices in $\mathcal{O}^{n}$. This
picture is closely related to the first picture. The third picture uses the
geometric Satake isomorphism and the interpretation of Littlewood-Richardson
coefficients as decomposition multiplicities for tensor products of finite
dimensional representation of $GL_{n}$, which are directly related to the
affine Grassmannian. The fourth picture is the number of irreducible
components of certain subvarieties of the Springer fibers in a generalized
flag manifold. The whole Section 3 is devoted to the discussion of the second
and the third picture. The fourth picture will be introduced in Section 4.
In this section we would like to discuss the link between module extensions
and geometry. Throughout this paper, we fix a ring
$\mathcal{O}=\mathbb{C}\left[\\!\left[t\right]\\!\right]$ with a maximal ideal
$\mathfrak{p}=t\mathcal{O}$. We will establish a geometric model for the
extensions of finite dimensional modules of the form

$M_{\lambda}=\mathbb{C}[t]/(t^{\lambda_{1}})\oplus\ldots\mathbb{C}[t]/(t^{\lambda_{n}})$

where $\lambda=\left(\lambda_{1}\ldots,\lambda_{n}\right)$ is a partition. We
can add zeros to the end of $\lambda$ without causing any change to
$M_{\lambda}$. The geometric model of module extensions is given by the
following lemma, which we will prove in the later sections of this paper:

######  Lemma 3.0.1.

For three partitions $\alpha,\beta,\gamma$ and three modules
$M_{\alpha},M_{\beta},M_{\gamma}$ as defined above, there exists an exact
sequence

$0\rightarrow M_{\alpha}\rightarrow M_{\gamma}\rightarrow M_{\beta}\rightarrow
0$

if and only if the partitions $\alpha,\beta,\gamma$ makes the Littlewood-
Richardson coefficient $c_{\alpha,\beta}^{\gamma}$ non-vanishing. The
Littlewood-Richardson coefficient $c_{\alpha,\beta}^{\gamma}$ is equal to the
number of isomorphism classes of all such possible extensions.

The first part of Theorem A is a direct consequence of this lemma.

###  3.1. Preliminaries on Affine Grassmannian

The _affine Grassmannian_ is a geometric object parametrizing the full-rank
$\mathcal{O}$-sublattices in $\mathcal{O}^{n}$.

######  Definition 3.1.1.

Setting $\mathcal{O}=\mathbb{C}\left[\\!\left[t\right]\\!\right]$ and
$K=\mathbb{C}\left(\\!\left(t\right)\\!\right)$, which are the rings of formal
power series and formal Laurent series, respectively, let $G$ be a semisimple
algebraic group, the _affine grassmannian_ of $G$ is defined as the quotient

$\mathrm{Gr}_{G}=G\left(K\right)/G\left(\mathcal{O}\right).$

The affine grassmannian is an _ind-scheme_ , which means that it is the direct
limit of finite-dimensional closed subschemes. To see this, we need a
stratification on the affine Grassmannian based on the _Cartan decomposition_
(See [MV07, Section 2] and [BR18, Proposition 1.3.2]), in which the cells are
parametrized by the dominant coweights

(1) $G(K)=\coprod_{\lambda\in
X_{*}\left(T\right)^{+}}G\left(\mathcal{O}\right)\cdot L_{\lambda}\cdot
G\left(\mathcal{O}\right).$

In the case that $G=GL_{n}$, $L_{\lambda}$ is the diagonal matrix
$L_{\lambda}=\mathrm{diag}\left(t^{\lambda_{1}},\ldots,t^{\lambda_{n}}\right)$
where $\lambda=(\lambda_{1},\ldots,\lambda_{n})$ is a partition of length $n$.
We define an _affine Schubert cell_ as the quotient

$\mathrm{Gr}_{G}^{\lambda}=G\left(\mathcal{O}\right)\cdot L_{\lambda}\cdot
G\left(\mathcal{O}\right)/G\left(\mathcal{O}\right).$

By [BR18, Proposition 1.3.2], the closure
$\overline{\mathrm{Gr}_{G}^{\lambda}}$ of this cell is the disjoint union of
affine Schubert cells $\mathrm{Gr}_{G}^{\mu}$ with $\mu\leq\lambda$:

$\overline{\mathrm{Gr}_{G}^{\lambda}}=\coprod_{\begin{subarray}{c}\mu\in
X_{*}\left(T\right)^{+}\\\
\mu\leq\lambda\end{subarray}}\mathrm{Gr}_{G}^{\mu}.$

In the disjoint union above, the order $\mu\leq\lambda$ on the partitions are
given by the closure order described in the beginning of Section 1.

###  3.2. Affine Schubert Varieties and Lattices in $\mathcal{O}^{n}$

We will use the affine Schubert cycles to parametrize full-rank
$\mathcal{O}$-sublattices in $\mathcal{O}^{n}$. The correspondence is
established by the following lemma:

######  Lemma 3.2.1.

For any dominant coweight $\lambda\in X_{*}(T)^{+}$, there is a bijection
between the elements $x\in\mathrm{Gr}_{G}^{\lambda}$ and the full-rank
$\mathcal{O}$-sublattices $\Lambda_{x}\subset\mathcal{O}^{n}$ such that

$\mathcal{O}^{n}/\Lambda_{x}\cong\bigoplus_{i}\mathbb{C}[t]/(t^{\lambda_{i}}).$

###### Proof.

Since each element $x\in\mathrm{Gr}_{G}^{\lambda}$ represents an element in
${G(\mathcal{O})}L_{\lambda}{G(\mathcal{O})}$ of the form

$x=\alpha L_{\lambda},\;\alpha\in G(\mathcal{O}),$

the column vectors of $L_{\lambda}$ generates a sublattice

$\alpha^{-1}\Lambda_{x}=t^{\lambda_{1}}\mathcal{O}\oplus\ldots\oplus
t^{\lambda_{n}}\mathcal{O}$

of $\alpha^{-1}\mathcal{O}^{n}\cong\mathcal{O}^{n}$. The quotient
$\alpha^{-1}\left(\mathcal{O}^{n}/\Lambda_{x}\right)\cong\mathcal{O}^{n}/\Lambda_{x}$
is a module which satisfies the requirement of the lemma. Conversely, if there
is a lattice $\Lambda_{x}=\bigoplus_{i}\mathcal{O}f_{i}$ which satisfies the
requirement of the lemma, the isomorphism will give an isomorphism between
lattices
$p:\bigoplus_{j}\mathcal{O}t^{\lambda_{j}}\rightarrow\Lambda_{x}=\bigoplus_{i}\mathcal{O}f_{i}$
such that $f_{j}=\sum_{i}r_{ji}t^{\lambda_{j}}$ where the matrix $(r_{ji})$ is
an invertible matrix in $G(\mathcal{O})$. ∎

As in the second picture mentioned in the beginning of Section 3.1, in order
to for us to describe the set of extensions of lattices, we need to introduce
the _twisted product_ $\mathrm{Gr}_{G}\widetilde{\times}\mathrm{Gr}_{G}$ of
two copies of the affine Grassmannians. Letting $G(\mathcal{O})$ act on
$G(\mathcal{O})\times G(\mathcal{O})$ by $k\cdot(a,b)=(ka,k^{-1}b)$, the
scheme $\mathrm{Gr}_{G}\widetilde{\times}\mathrm{Gr}_{G}$ is defined as the
quotient of $G(K)\times\mathrm{Gr}_{G}$ by the action of $G(\mathcal{O})$ on
$G(\mathcal{O})\times G(\mathcal{O})$, fitting into the diagram

(2)
$\textstyle{G(K)\times\mathrm{Gr}_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{\mathrm{Gr}_{G}\times\mathrm{Gr}_{G}}$$\textstyle{\mathrm{Gr}_{G}\widetilde{\times}\mathrm{Gr}_{G}}$

where the morphism $p$ is the quotient by $G(\mathcal{O})$ from the first
component. There is a _semi-small_ , ind _-proper_ multiplication map
$m:\mathrm{Gr}_{G}\tilde{\times}\mathrm{Gr}_{G}\rightarrow\mathrm{Gr}_{G}$
defined by

$m:(g,x)\mapsto gx.$

The reason why this map $m$ is called the multiplication map will be explained
in Lemma 3.2.4.
The twisted product $\widetilde{\times}$ can be defined similarly for the
Schubert cells and Schubert varieties by simply taking the quotient of the
direct product of the corresponding cells by $G(\mathcal{O})$. When restricted
to the twisted product of the Schubert varieties, the multiplication map $m$
maps
$\overline{\mathrm{Gr}_{G}^{\alpha}\widetilde{\times}\mathrm{Gr}_{G}^{\beta}}$
surjectively onto $\overline{\mathrm{Gr}_{G}^{\alpha+\beta}}$.
We can also take the twisted product of multiple affine Grassmannians and
Schubert varieties. The variety
$\mathrm{Gr}_{G}^{\lambda^{1},\ldots,\lambda^{n}}$ is defined as the multiple
twisted product

$\mathrm{Gr}_{G}^{\lambda^{1},\ldots,\lambda^{n}}=\mathrm{Gr}_{G}^{\lambda^{1}}\widetilde{\times}\ldots\widetilde{\times}\mathrm{Gr}_{G}^{\lambda^{n}}.$

We can stratify the space
$\overline{\mathrm{Gr}_{G}^{\lambda^{1},\ldots,\lambda^{n}}}$ by the twisted
product of Schubert cells corresponding to smaller partitions:

$\overline{\mathrm{Gr}_{G}^{\lambda^{1},\ldots,\lambda^{n}}}=\bigcup_{\mu^{i}\leq\lambda^{i}}\mathrm{Gr}_{G}^{\mu^{1},\ldots,\mu^{n}}.$

To count the dimension of the twisted product of these Schubert cells, we need
the following lemma from [Zhu16] and [BR18]:

######  Lemma 3.2.2.

The dimension of the twisted product of cells
$\mathrm{Gr}_{G}^{\lambda^{1},\ldots,\lambda^{n}}$ is equal to $\langle
2\rho,\sum_{i=1}^{n}\lambda^{i}\rangle$. The dimension of the fiber
$m^{-1}(x)\cap\mathrm{Gr}_{G}^{\lambda^{1},\ldots,\lambda^{n}}$ of $m$ over
any generic point $x\in\mathrm{Gr}_{G}^{\lambda}\subset\mathrm{Gr}_{G}$ is
less than or equal to $\langle\rho,\sum_{i=1}^{n}\lambda^{i}-\lambda\rangle$,
hence $m$ is a semi-small map.

###### Proof.

Since the stablizer of the action of $G(\mathcal{O})$ at $L^{\lambda}$ is
$G(\mathcal{O})\cap L_{\lambda}G(\mathcal{O})(L_{\lambda})^{-1}$, thus the
tangent space $\mathrm{Gr}_{G}^{\lambda}$ at $L_{\lambda}$ is

$\mathfrak{g}(\mathcal{O})/\mathfrak{g}(\mathcal{O})\cap\mathrm{Ad}_{L_{\lambda}}\mathfrak{g}(\mathcal{O})\cong\bigoplus_{\langle\alpha,\lambda\rangle\geq
0}\mathfrak{g}_{\alpha}(\mathcal{O})/t^{\langle\alpha,\lambda\rangle}\mathfrak{g}(\mathcal{O}).$

The dimension of this tangent space above $\langle 2\rho,\lambda\rangle$.
The dimension of the twisted product follows from a similar stabilizer
argument. The inequality for the dimension of the fiber
$m^{-1}(x)\cap\mathrm{Gr}_{G}^{\lambda^{1},\ldots,\lambda^{n}}$ follows from
[MV07, Lemma 4.4], also see [BR18, Lemma 6.4]. ∎

######  Remark 3.2.3.

For any partition $\lambda$, the number $\langle\rho,\lambda\rangle$ is
closely related to
$\frac{1}{2}\left\|\lambda^{t}\right\|^{2}=\frac{1}{2}\sum\left(\lambda^{t}_{i}\right)^{2}$.
Since we know

$\lambda_{i}-\lambda_{i+1}=\\#\left|\left\\{j\mid\lambda^{t}_{j}=i\right\\}\right|,$

we can express $\frac{1}{2}\left\|\lambda^{t}\right\|^{2}$ in terms of
$\lambda$:

$\frac{1}{2}\left\|\lambda^{t}\right\|^{2}=\sum_{i=1}i^{2}(\lambda_{i}-\lambda_{i+1}).$

Through simple calculation, we see that

$\frac{1}{2}\left\|\lambda^{t}\right\|^{2}+\langle\rho,\lambda\rangle=\left(n-\frac{1}{2}\right)\sum_{k=1}^{n}\lambda_{k}.$

Therefore, we can express the dimension of the fiber
$m^{-1}(x)\cap\mathrm{Gr}_{G}^{\lambda^{1},\ldots,\lambda^{n}}$ differently in
terms of the conjugate partitions:

(3)
$\dim\left(m^{-1}(x)\cap\mathrm{Gr}_{G}^{\lambda^{1},\ldots,\lambda^{n}}\right)=\frac{1}{2}\left(\left\|\lambda^{t}\right\|^{2}-\sum_{i}\left\|\lambda_{i}^{t}\right\|^{2}\right).$

We can use the variety
$\mathrm{Gr}_{G}^{\alpha}\widetilde{\times}\mathrm{Gr}_{G}^{\beta}$ to
parametrize the extensions of lattices (as promised in our second picture).
This fact is shown in the following lemma:

######  Lemma 3.2.4.

For any lattice $\Lambda\subset\mathcal{O}^{n}$, denote by
$\widehat{\Lambda}=\mathrm{Hom}_{\mathcal{\mathcal{O}}}(\Lambda,\mathcal{O})$
its dual lattice. The variety
$\mathrm{Gr}_{G}^{\alpha}\tilde{\times}\mathrm{Gr}_{G}^{\beta}$ parametrizes
the collection of sublattices
$\Lambda_{\gamma}\subset\Lambda_{\alpha}\subset\mathcal{O}^{n}$, such that

  1. (1)

They both span the same vector space $K^{n}$ over $K=\mathbb{C}(\\!(t)\\!)$,

  2. (2)

$\widehat{\Lambda_{\gamma}}/\mathcal{O}^{n}\cong M_{\gamma}$, and
$\widehat{\Lambda_{\alpha}}/\mathcal{O}^{n}\cong M_{\alpha}$,

  3. (3)

$\Lambda_{\alpha}/\Lambda_{\gamma}\cong M_{\beta}$.

The multiplication map
$m:\overline{\mathrm{Gr}_{G}^{\alpha}\tilde{\times}\mathrm{Gr}_{G}^{\beta}}\rightarrow\overline{\mathrm{Gr}_{G}^{\alpha+\beta}}$
sends any pair of such lattices $(\Lambda_{\gamma},\Lambda_{\alpha})$ to
$\Lambda_{\gamma}$.

###### Proof.

The scheme $\mathrm{Gr}_{G}^{\alpha}\widetilde{\times}\mathrm{Gr}_{G}^{\beta}$
is a $G(\mathcal{O})$-quotient of the product
$G(\mathcal{O})L_{\alpha}G(\mathcal{O})\times G(\mathcal{O})L_{\beta}$, which
parametrizes pairs of elements $(l_{1},l_{2})$ with $l_{2}\in
G(\mathcal{O})L_{\beta}$ and $l_{1}\in
G(\mathcal{O})L_{\alpha}G(\mathcal{O})$. Each equivalent class under the
quotient by $G(\mathcal{O})$ has a unique representative of the form
$(xL_{\alpha},yL_{\beta})$, and can be moved to a representative
$(L_{\alpha},x^{-1}yL_{\beta})$. In the moved lattice $x^{-1}\mathcal{O}^{n}$,
one can consider the lattice generated by the column vectors of the matrix
$L_{\alpha}x^{-1}yL_{\beta}$. We would like to prove that this lattice is the
correct $x\Lambda_{\gamma}$,

$L_{\alpha}=(e_{1},\ldots,e_{n}),$

where each $e_{i}=(0,\ldots,t^{\alpha_{i}},\ldots,0)^{t}$ is only nonzero at
the $i$-th entry. We represent by a matrix $x^{-1}yL_{\beta}=(g_{i,j})$, and
column vectors of the matrix $L_{\alpha}x^{-1}yL_{\beta}$ are

$v_{j}=(t^{\alpha_{i}}g_{i,j})=\sum_{i}g_{i,j}e_{i}$

which generates a sublattice $x\Lambda_{\gamma}$ of $\sum_{i}\mathcal{O}e_{i}$
which satisfies $\Lambda_{\alpha}/\Lambda_{\gamma}\cong M_{\beta}$. ∎

By Lemma 3.2.4, the two projections $\pi_{1},\pi_{2}$ can be described as maps
from the twisted product $\mathrm{Gr}_{G}\widetilde{\times}\mathrm{Gr}_{G}$
onto the first and the second factor, respectively. The projection $\pi_{1}$
sends each pair of $(\Lambda_{\gamma},\Lambda_{\alpha})$ to
$\Lambda_{\alpha}$, while the projection $\pi_{2}$ sends the pair
$(\Lambda_{\gamma},\Lambda_{\alpha})$ to the sublattice
$\Lambda_{\beta}=L_{\alpha}^{-1}x^{-1}\Lambda_{\gamma}$ of $\mathcal{O}^{n}$.
These two projections can also be used to set up an isomorphism between the
twisted product $\mathrm{Gr}_{G}\tilde{\times}\mathrm{Gr}_{G}$ and the direct
product $\mathrm{Gr}_{G}\times\mathrm{Gr}_{G}$, which sends any element $y$ in
the twisted product to $(\pi_{1}(y),\pi_{2}(y))$. In terms of group element
representatives, the isomorphism can be described by the map

(4) $\displaystyle\mathrm{Gr}_{G}\times\mathrm{Gr}_{G}$
$\displaystyle\longrightarrow\mathrm{Gr}_{G}\tilde{\times}\mathrm{Gr}_{G}$
$\displaystyle(x_{1},x_{2})$
$\displaystyle\longmapsto(x_{1},x_{1}^{-1}x_{2}).$

###  3.3. Equivariant Perverse Sheaf and the Geometric Satake Equivalence

According to [BL06, Theorem 2.6.2], for any $G(\mathcal{O})\times
G(\mathcal{O})$-equivariant constructible sheaf $\mathcal{F}$ on
$G(K)\times\mathrm{Gr}_{G}$, there is an unique $G(\mathcal{O})$-equivariant
constructible sheaf $\widetilde{\mathcal{F}}$ on
$\mathrm{Gr}_{G}\tilde{\times}\mathrm{Gr}_{G}$ such that
$q^{*}\widetilde{\mathcal{F}}=\mathcal{F}$. In particular, for two
constructible sheaves $\mathcal{A,B}$ on $\mathrm{Gr}_{G}$, consider the
pullback $p^{*}(\pi_{1}^{*}\mathcal{A}\boxtimes\pi_{2}^{*}\mathcal{B})$ as the
sheaf $\mathcal{F}$, where $\pi_{1},\pi_{2}$ are the projection to the first
and second factors defined in the previous section, respectively. We denote
the corresponding unique constructible sheaf $\widetilde{\mathcal{F}}$ by
$\pi_{1}^{*}\mathcal{A}\widetilde{\boxtimes}\pi_{2}^{*}\mathcal{B}$. We can
thus define a _convolution product_ in the derived category of
$G(\mathcal{O})$-equivariant constructible sheaves on $\mathrm{Gr_{G}}$ as

$\mathcal{A}\star\mathcal{B}=Rm_{!}\left(\pi_{1}^{*}\mathcal{A}\widetilde{\boxtimes}\pi_{2}^{*}\mathcal{B}\right).$

Equipped with the convolution product, the category
$\mathrm{Perv}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ of
$G(\mathcal{O})$-equivariant perverse sheaves is a monoidal category with
irreducible objects the intersection cohomology sheaves

$\mathrm{IC}^{\lambda}:=\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\lambda}},{\mathbb{C}})=\iota^{\lambda}_{!*}\left({\mathbb{C}}[\dim\mathrm{Gr}^{\lambda}_{G}]\right)$

supported on each Schubert variety $\overline{\mathrm{Gr}_{G}^{\lambda}}$,
which are the intermediate extensions of the locally trivial bundle of
$\mathrm{Gr}_{G}^{\lambda}$ along the inclusion
$\iota:\mathrm{Gr}_{G}^{\lambda}\hookrightarrow\overline{\mathrm{Gr}_{G}^{\lambda}}$.
The main takeaway in the proof of our main Lemma 3.0.1 is the calculation of
the convolution product
$\mathrm{IC}^{\lambda_{1}}\star\mathrm{IC}^{\lambda_{2}}$ of the two
intersection cohomology sheaves. The main reference of the equivariant
perverse sheaves on the affine Grassmannian in geneal is [BR18, Section 7.5].

The _geometric_ _Satake_ _equivalence_ (see [Gin95],[MV07] or [BR18]) is an
equivalence of monoidal categories between the category of
$G(\mathcal{O})$-equivariant perverse sheaves
$\mathrm{Perv}_{G\left(\mathcal{O}\right)}\left(\mathrm{Gr}_{G}\right)$ and
the category of finite dimensional representations
$\mathrm{Rep}_{\mathbb{C}}\left(\check{G}\right)$ of the Langlands dual group
$\check{G}$:

$\mathrm{Perv}_{G\left(\mathcal{O}\right)}\left(\mathrm{Gr}_{G}\right)\cong\mathrm{Rep}_{\mathbb{C}}\left(\check{G}\right).$

The actual construction of the categorical equivalence is not trivial, and
requires the Tannakaian reconstruction theorems for an actual group scheme
$\check{G}$ which makes the following diagram of functors commute:

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This commutative diagram of functors respects the tensor products and the
units of the three tensor categories. The total cohomology functor $H^{*}$ is
the graded cohomology which decomposes into weight components:

$H^{k}(\mathrm{Gr}_{G},\mathcal{A})\cong\bigoplus_{\begin{subarray}{c}\mu\in
X_{*}(T)\\\ \langle 2\rho,\mu\rangle=k\end{subarray}}H^{\langle
2\rho,\mu\rangle}(T_{\mu},\iota^{!}\mathcal{A})$

where $T_{\mu}$ is the locally closed subvariety of $\mathrm{Gr}_{G}$
consisting of the points whose limit at $\infty$ is $L_{\mu}$ under the
adjoint action of a regular dominant coroot.

If we identify the cocharacter lattice $X_{*}(T)$, which parametrizes the
irreducible objects of the category
$\mathrm{Perv}_{G\left(\mathcal{O}\right)}\left(\mathrm{Gr}_{G}\right)$, with
the character lattice $X^{*}(\check{T})$ of the Cartan subgroup in the
Langlands dual group, the functor $S$ sends the intersection cohomology sheaf
supported on $\overline{\mathrm{Gr}_{G}^{\lambda}}$ for each dominant coweight
$\lambda$ to the irreducible representation $V(\lambda)$ of $\check{G}$ with
highest weight $\lambda$:

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and the convolution product of the intersection cohomology sheaves

$\mathrm{IC}^{\lambda^{1}}\star\ldots\star\mathrm{IC}^{\lambda^{n}}=Rm_{!}\left(\pi_{1}^{*}\mathrm{IC}^{\lambda^{1}}\tilde{\boxtimes}\ldots\tilde{\boxtimes}\pi_{n}^{*}\mathrm{IC}^{\lambda^{n}}\right)$

is sent to the tensor product $V\left(\lambda^{1}\right)\otimes\ldots\otimes
V\left(\lambda^{n}\right)$. The functor $Rm_{!}$, which can be identified as
$Rm_{*}$ due to the properness of $m$, sends a complex of constructible
sheaves
$\pi_{1}^{*}\mathrm{IC}^{\lambda^{1}}\tilde{\boxtimes}\ldots\tilde{\boxtimes}\pi_{n}^{*}\mathrm{IC}^{\lambda^{n}}$
on $\overline{\mathrm{Gr}_{G}^{\lambda^{1},\ldots,\lambda^{n}}}$ to a complex
of constructible sheaves on
$\overline{\mathrm{Gr}_{G}^{\lambda^{1}+\ldots+\lambda^{n}}}$. As a derived
direct image, the constructible sheaf
$\mathrm{IC}^{\lambda^{1}}\star\ldots\star\mathrm{IC}^{\lambda^{n}}$ is
perverse due to the semi-smallness of the morphism $m$. We need a lemma for
use in the next section:

######  Lemma 3.3.1.

The twisted product
$\pi_{1}^{*}\mathrm{IC}^{\lambda^{1}}\tilde{\boxtimes}\pi_{2}^{*}\mathrm{IC}^{\lambda^{2}}$
is isomorphic to the intersection cohomology sheaf
$\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\lambda^{1},\lambda^{2}}})$.

###### Proof.

Consider the following pull-back square

$\textstyle{{\overline{\mathrm{Gr}_{G}^{\lambda^{1}}\times\mathrm{Gr}_{G}^{\lambda^{2}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\pi_{2}}$$\textstyle{{\overline{\mathrm{Gr}_{G}^{\lambda^{1},\lambda^{2}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{\overline{\mathrm{Gr}_{G}^{\lambda^{1}+\ldots+\lambda^{n}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathrm{Gr}_{G}^{\lambda^{1}+\ldots+\lambda^{n}}}}$

such that the top row isomorphism $h$ sends $(x_{1},x_{2})$ to
$(x_{1},x_{1}^{-1}x_{2})$ as described in the final remarks of Section 3.2.
The morphism $h$ is an isomorphism which preserves the strata of
$\overline{\mathrm{Gr}_{G}^{\lambda^{1}}\times\mathrm{Gr}_{G}^{\lambda^{2}}}$
and ${\overline{\mathrm{Gr}_{G}^{\lambda^{1},\lambda^{2}}}}$, and hence sends
the external tensor product of IC-sheaves

$\pi_{1}^{*}\mathrm{IC}^{\lambda^{1}}\boxtimes\pi_{2}^{*}\mathrm{IC}^{\lambda^{2}}\cong\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\lambda^{1}}\times\mathrm{Gr}_{G}^{\lambda^{2}}})$

to the twisted tensor product
$\pi_{1}^{*}\mathrm{IC}^{\lambda^{1}}\widetilde{\boxtimes}\pi_{2}^{*}\mathrm{IC}^{\lambda^{2}}$.
Therefore, the twisted tensor product is also the IC-sheaf
$\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\lambda_{1},\lambda_{2}}})$. ∎

The fact that $S$ is an equivalence of tensor categories identifies the
$\hom$-sets between the intersection cohomology sheaves and those between the
finite dimensional representations:

$\mathrm{Hom}(\mathrm{IC}^{\gamma},\mathrm{IC}^{\alpha}\star\mathrm{IC}^{\beta})\cong\mathrm{Hom}_{G}(V(\gamma),V(\alpha)\otimes
V(\beta)).$

The dimension of this space is equal to the Littlewood-Richardson coefficient
$c_{\alpha,\beta}^{\gamma}$.

###  3.4. The Decomposition Theorem and the Proof of the Lemma 3.0.1

The derived direct image
$Rm_{*}\mathrm{IC}\left(\overline{\mathrm{Gr}_{G}^{\lambda^{1},\ldots,\lambda^{n}}}\right)$
of the intersection cohomology sheaf on the variety
$\overline{\mathrm{Gr}_{G}^{\lambda^{1},\ldots,\lambda^{n}}}$ along the semi-
small map
$m:\mathrm{Gr}_{G}\widetilde{\times}\mathrm{Gr}_{G}\rightarrow\mathrm{Gr}_{G}$
can be decomposed non-canonically as a direct sum of irreducible objects. See
[dCM09, Theorem 1.6.1] for the following well-known decomposition theorem:

######  Theorem 3.4.1 (Beĭlinson-Bernstein-Deligne-Gabber).

Let $f:X\rightarrow Y$ be a proper map of algebraic varieties, the derived
push-forward $Rf_{*}$ of an intersection cohomology sheaf $\mathrm{IC}_{X}$
decomposes (noncanonically) into a direct sum of semisimple perverse sheaves

$Rf_{*}\mathrm{IC}_{X}\cong\bigoplus_{i\in\mathbb{Z}}{}^{p}\mathcal{H}^{i}\left(Rf_{*}\mathrm{IC}_{X}\right)[-i].$

There is a canonical decomposition of
${}^{p}\mathcal{H}^{i}\left(Rf_{*}\mathrm{IC}_{X}\right)$ as a direct sum of
intersection cohomology sheaves on $Y$, where ${}^{p}\mathcal{H}^{i}$ is the
perverse cohomology functor ${}^{p}\tau_{\leq i}\circ{}^{p}\tau^{\geq
i}={}^{p}\tau^{\geq i}\circ{}^{p}\tau_{\leq i}$.

We will make use of the decomposition theorem in the form of its corollary
(see also [Hai03, Lemma 3.2]):

######  Corollary 3.4.2.

The convolution product of IC-sheaves decomposes into a direct sum

$\mathrm{IC}^{\alpha}\star\mathrm{IC}^{\beta}=\bigoplus_{\gamma\leq\alpha+\beta}L_{\alpha,\beta}^{\gamma}\otimes\mathrm{IC}^{\gamma}$

where $L_{\alpha,\beta}^{\gamma}=H^{\langle
2\rho,\alpha+\beta-\gamma\rangle}_{c}(m^{-1}(x)\cap\mathrm{Gr}^{\alpha,\beta}_{G})$
for a generic $x\in\mathrm{Gr}_{G}^{\gamma}$.

###### Proof.

We consider embeddings of the big cell
$\mathrm{Gr}_{G}^{\alpha,\beta}\subset\overline{\mathrm{Gr}_{G}^{\alpha,\beta}}$
and its complement $Z^{\alpha,\beta}$ in
$\overline{\mathrm{Gr}_{G}^{\alpha,\beta}}$:

$\textstyle{Z^{\alpha,\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\overline{\mathrm{Gr}_{G}^{\alpha,\beta}}}$$\textstyle{\mathrm{Gr}_{G}^{\alpha,\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$

and the corresponding embeddings of their intersections with the fiber of $m$:

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The stalk of the cohomology sheaf
${}^{p}\mathcal{H}^{i}\left(Rm_{*}\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}})\right)$
at $x\in\mathrm{Gr}_{G}^{\gamma}$ can be identified as the cohomology group
$H^{i}(m^{-1}(x),\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}})\mid_{m^{-1}(x)})$.
To compute this cohomology group, we need to apply the cohomology functor to
the distinguished triangle

$j_{!}j^{*}\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}})\rightarrow\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}})\rightarrow
i_{!}i^{*}\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}})\rightarrow$

which gives rise to a long exact sequence of cohomologies

(5) $\displaystyle\ldots\rightarrow
H_{c}^{i}(m^{-1}(x)\cap\mathrm{Gr}_{G}^{\alpha,\beta},\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}}))\rightarrow
H^{i}(m^{-1}(x),\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}}))\rightarrow
H_{c}^{i}(m^{-1}(x)\cap
Z^{\alpha,\beta},\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}}))\rightarrow\ldots.$

Since $\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}})$ is a perverse
sheaf, the cohomology sheaf
$\mathcal{H}^{i}(i_{\gamma}^{*}\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}}))=0$
if $i\geq-\langle 2\rho,\alpha+\beta\rangle$. Also since the hypercohomology
spectral sequence converges:

$\displaystyle E_{2}^{pq}:H^{p}(m^{-1}(x)\cap
Z^{\alpha,\beta},\mathcal{H}^{q}\left(\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}})\right)\implies
H^{p+q}(m^{-1}(x)\cap
Z^{\alpha,\beta},i^{*}_{\gamma}\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}})),$

all the nonzero terms in this spectral sequence appears when

$q<-\langle 2\rho,\alpha+\beta\rangle,\;p\leq\dim(m^{-1}(x)\cap
Z^{\alpha,\beta})\leq\langle 2\rho,\alpha+\beta-\gamma\rangle-2.$

Therefore, $H_{c}^{i}(m^{-1}(x)\cap
Z^{\alpha,\beta},i^{*}_{\gamma}\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}}))$
can be possibly nonzero only when $i\leq-\langle 2\rho,\gamma\rangle-2$. Again
by the long exact sequence (5), the stalk of the cohomology sheaf

$H^{-\langle
2\rho,\gamma\rangle}(m^{-1}(x),\mathrm{IC}(\overline{\mathrm{Gr}^{\alpha,\beta}_{G}}))\cong
H_{c}^{-\langle
2\rho,\gamma\rangle}(m^{-1}(x)\cap\mathrm{Gr}_{G}^{\alpha,\beta},j^{*}_{\gamma}\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}})).$

When restricted to the open cell,
$\mathrm{IC}(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}})$ is the constant local
system $\mathbb{C}[\langle 2\rho,\alpha+\beta\rangle]$. Therefore, the stalk
of the cohomology sheaf at $x$ is isomorphic to

$H^{-\langle
2\rho,\gamma\rangle}(m^{-1}(x),\mathrm{IC}(\overline{\mathrm{Gr}^{\alpha,\beta}_{G}}))\cong
H^{\langle
2\rho,\alpha+\beta-\gamma\rangle}_{c}(m^{-1}(x)\cap\mathrm{Gr}^{\alpha,\beta}_{G}).$

We denote this space by $L_{\alpha,\beta}^{\gamma}$. This finishes the proof
of the corollary. ∎

Now we are ready to prove Lemma 3.0.1.

###### Proof of Lemma 3.0.1.

By Lemma 3.2.1, any element $x\in\mathrm{Gr}_{G}^{\gamma}$ represents a
lattice $\Lambda_{x}$ such that the quotient $\mathcal{O}^{n}/\Lambda_{x}$ has
Jordan type $\gamma$. If the multiplication map

$m:\mathrm{Gr}_{G}^{\alpha}\widetilde{\times}\mathrm{Gr}_{G}^{\beta}\rightarrow\overline{\mathrm{Gr}_{G}^{\alpha+\beta}}$

sends a pair $y=(\Lambda_{\gamma},\Lambda_{\alpha})$ to $\Lambda_{x}$, by
Lemma 3.2.4 we must have $\Lambda_{\gamma}=\Lambda_{x}$. Therefore, in order
to prove Lemma 3.0.1, we will have to show that for any
$x\in\mathrm{Gr}_{G}^{\gamma}$, the fiber
$m^{-1}(x)\cap\mathrm{Gr}_{G}^{\alpha,\beta}$ is nonempty if and only if
$c_{\alpha,\beta}^{\gamma}=\dim L_{\alpha,\beta}^{\gamma}\neq 0$.
By the decomposition theorem and the semi-simplicity of the category of
equivariant perverse sheaves,
$\mathrm{Hom}(\mathrm{IC}^{\gamma},\mathrm{IC}^{\alpha}\star\mathrm{IC}^{\beta})$
can be decomposed noncanonically as a direct sum:

$\displaystyle\mathrm{Hom}(\mathrm{IC}^{\gamma},\mathrm{IC}^{\alpha}\star\mathrm{IC}^{\beta})$
$\displaystyle\cong\mathrm{Hom}(\mathrm{IC}^{\gamma},Rm_{*}\mathrm{IC}\left(\overline{\mathrm{Gr}_{G}^{\alpha,\beta}}\right))$
$\displaystyle=\mathrm{Hom}\left(\mathrm{IC}^{\gamma},\bigoplus_{\begin{subarray}{c}\mathrm{Gr}^{\sigma}_{G}\subset\overline{x\in\mathrm{Gr}^{\alpha+\beta}_{G}}\\\
m^{-1}(x)\cap\mathrm{Gr}^{\alpha,\beta}_{G}\neq\emptyset\end{subarray}}L_{\alpha,\beta}^{\sigma}\otimes\mathrm{IC}^{\sigma}\right)$
$\displaystyle=L^{\gamma}_{\alpha,\beta}.$

By the geometric Satake equivalence, the dimension of the space is equal to
the multiplicity of $V(\gamma)$ in $V(\alpha)\otimes V(\beta)$, which is equal
to the Littlewood–Richardson coefficient $c^{\gamma}_{\alpha,\beta}$.
Since on each Schubert cycle $\mathrm{Gr}_{G}^{\gamma}$, the group $G(K)$ acts
transitively, the preimage of the Schubert cycle
$m^{-1}(\mathrm{Gr}_{G}^{\gamma})$ intersects $\mathrm{Gr}^{\alpha,\beta}_{G}$
if and only if the fiber $m^{-1}(x)$ intersects
$\mathrm{Gr}^{\alpha,\beta}_{G}$. By [Zhu16, Corollary 5.1.5], the dimension
of the space

$L^{\gamma}_{\alpha,\beta}=H^{\langle
2\rho,\alpha+\beta-\gamma\rangle}_{c}(m^{-1}(x)\cap\mathrm{Gr}^{\alpha,\beta}_{G}).$

is the number of irreducible components of
$m^{-1}(x)\cap\mathrm{Gr}^{\alpha,\beta}_{G}$ of complex dimension
$\langle\rho,\alpha+\beta-\gamma\rangle$. All the dominant coweights of
$GL_{n}$ are sums of coweights of the form
$\epsilon^{i}=\left(1^{i},0^{n-i}\right)$, which are all _minuscule_ , i.e.
$\langle\epsilon_{i},\alpha\rangle\in\\{-1,0,1\\}$ for any root $\alpha$. By
[Hai06, Theorem 1.3], the irreducible components of
$m^{-1}(x)\cap\mathrm{Gr}^{\alpha,\beta}_{G}$, if not empty, are
equidimensional and of dimension $\langle\rho,\alpha+\beta-\gamma\rangle$.
Since the Littlewood-Richardson coefficient $c_{\alpha,\beta}^{\gamma}$ is
equal to the number of irreducible components of
$m^{-1}(x)\cap\mathrm{Gr}^{\alpha,\beta}_{G}$, we have now proven that
$m^{-1}(x)\cap\mathrm{Gr}^{\alpha,\beta}_{G}\neq\emptyset$ if and only if
$c_{\alpha,\beta}^{\gamma}\neq 0$. ∎

##  4\. Interpretations of the Littlewood-Richardson Coefficient

In this section, we consider the Young diagrams with shapes
$\alpha,\beta\subset\gamma$ such that $c_{\alpha,\beta}^{\gamma}\neq 0$. We
will fix our notations to denote the length of $\gamma$ by $m$, the length of
$\alpha$ by $n$. The goal of this section is to prove the second part of the
Theorem A, which follows from the geometric interpretation of the Littlewood-
Richardson coefficients studied by Springer in [Spr78] and Marc van Leeuwen in
[vL00]. This geometric interpretation of $c_{\alpha,\beta}^{\gamma}$ is our
fourth picture mentioned in the beginning of Section 3.1.

###  4.1. The Satake Fiber

The fiber
$\mathcal{S}^{\gamma}_{\alpha,\beta}=m^{-1}(L_{\gamma})\cap\mathrm{Gr}^{\alpha,\beta}_{G}$
of the multiplication map $m$ above the point
$L_{\gamma}\in\mathrm{Gr}_{\gamma}$ is sometimes referred to as the _Satake
fiber_. The following lemma identifies the Satake fiber as a subvariety of the
partial flag variety $G/P_{[n,m-n]}$:

######  Lemma 4.1.1.

The Satake fiber $\mathcal{S}^{\gamma}_{\alpha,\beta}$ parametrizes the
subspaces $V$ of dimension $|\alpha|$ inside a vector space $W$ of dimension
$|\alpha|+|\beta|$, such that there is a nilpotent element $x$ which stablizes
$V\subset W$, and

  1. (1)

$x\mid_{V}$ acts with Jordan type $\alpha$,

  2. (2)

$x$ acts on $W/V$ with Jordan type $\beta$,

  3. (3)

$x$ acts on $W$ with Jordan type $\gamma$.

###### Proof.

The Satake fiber $m^{-1}(L_{\gamma})\cap\mathrm{Gr}^{\alpha,\beta}_{G}$
parametrizes the pairs of lattices $(\Lambda_{\gamma},\Lambda_{\alpha})$
satisfying the conditions in Lemma 3.2.4. The quotients
$M_{\alpha},M_{\beta},M_{\gamma}$ of $\mathcal{O}^{n}$ by the three lattices
$\Lambda_{\alpha},\Lambda_{\beta}$ and $\Lambda_{\gamma}$, respectively, fit
into the exact sequence

$0\rightarrow M_{\alpha}\rightarrow M_{\gamma}\rightarrow M_{\beta}\rightarrow
0.$

The multiplication by the variable $t$ of the ring $\mathcal{O}[\\![t]\\!]$ is
a nilpotent linear operator with Jordan type $\alpha,\beta,\gamma$ on the
three spaces, respectively. Conversely, if we have a pair of spaces $V\subset
W$ satisfying the conditions of this lemma, $V$ and $W$ can be viewed as the
$\mathcal{O}$-modules $M_{\alpha},M_{\gamma}$, respectively. ∎

###  4.2. The Springer-type Fibers

The tangent bundle of the flag variety $G/B$ can be realized as a subset of
the product of the nilpotent cone $\mathcal{N}$ and $\mathcal{B}=G/B$:

$T^{*}\mathcal{B}=\left\\{(x,gB)\mid
x\in\mathrm{Ad}(g)\mathfrak{b}\right\\}\subset\mathcal{N}\times\mathcal{B}.$

For each nilpotent orbit
$\mathcal{O}_{\gamma}\subset\mathcal{N}\subset\mathfrak{g}$, the projection
$p$ from the cotangent bundle $T^{*}\mathcal{B}$ to $\mathcal{B}$ is a
resolution of singularities called the _Springer resolution_ for the nilpotent
cone $\mathcal{N}$:

$\textstyle{T^{*}\mathcal{B}\subset\mathcal{N}\times\mathcal{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{p}$$\textstyle{\mathcal{N}}$$\textstyle{\mathcal{B}}$

where $\pi,p$ are the projections onto the first and the second factor,
respectively. The _Springer fiber_ $\mathcal{B}_{\gamma}(x)$ above
$x\in\mathcal{O}_{\gamma}$ of $p$ is the collection of Borel subalgebras
$\mathfrak{b}\subset\mathfrak{g}$ containing the nilpotent element $x$

$\mathcal{B}_{\gamma}(x)=\left\\{(x,gB)\mid
x\in\mathrm{Ad}(g)\mathfrak{b}\right\\}.$

It is proven by Spaltenstein in [Spa76] and Steinberg in [Ste76] and [Ste88]
that the irreducible components of the Springer fiber
$\mathcal{B}_{\gamma}(x)$ are equidimensional and are parametrized by the
_standard Young tableaux_ of shape $\gamma$. The number of standard Young
tableaux can be calculated from the hook-length formula.
For $G=GL_{m}$, the Springer fiber $\mathcal{B}_{\gamma}(x)$ can be
interpreted as the collection of $x$-stable full flags

$\\{0\\}=V_{0}\subset V_{1}\subset V_{2}\subset\ldots\subset V_{m}=V$

in a vector space $V_{m}$ of dimension $m$, on which $x$ acts as a nilpotent
linear transformation. Since the Springer fibers are isomorphic for different
choices of $x$ in the same orbit $\mathcal{O}_{\mu}$, we can denote the
Springer fiber by $\mathcal{B}_{\gamma}$ if there is no ambiguity caused by
different choices of $x$.

Apart from the case of the full flag variety $G/B$, we can also consider the
Springer-type partial resolution from the cotangent bundle of a partial flag
variety $G/P$ corresponding to the parabolic subgroup $P=LN$. Similar to the
Springer resolution of the nilpotent cone by the cotangent bundles of a full
flag variety, given a parabolic subgroup $P$, we can define a _partial
resolution_ $\widetilde{\mathcal{N}_{P}}$ of the nilpotent cone $\mathcal{N}$

$\widetilde{\mathcal{N}_{P}}=\left\\{(x,gP)\mid
x\in\mathcal{N}\cap\mathrm{Ad}(g)\mathfrak{p}\right\\}\subset\mathcal{N}\times
G/P\xrightarrow{\pi_{P}}\mathcal{N}$

which factorizes the Springer resolution $\pi$. The _generalized Springer
fiber_ $\pi_{P}^{-1}(x)$ above any element $x$ is sometimes referred to as the
_Steinberg variety_ (cf. [BM, Chapter 3], some references also call it the
_Spaltenstein variety_). When $P=B$, the partial resolution $\pi_{P}$ becomes
the usual Springer resolution, and the fiber $\pi^{-1}_{P}(x)$ becomes the
usual Springer fiber above $x$. Equivalently, if we consider the parabolic
subgroup $P$ as the stabilizer of a partial flag

$V_{k_{1}}\subset V_{k_{2}}\subset\ldots\subset V_{k_{r}}=V_{m}$

where each $V_{k_{i}}$ is a subspace of dimension $k_{i}$. The fiber
$\pi_{P}^{-1}(x)$ can be identified as the collection of the $x$-stable flags
which are stabilized by a conjugate of $P$.

We are particularly interested in the standard parabolic subgroups
$P_{[n,1^{m-n}]}$ with a Levi subgroup $L_{[n,1^{m-n}]}\cong GL(n)\times
GL(1)^{m-n}$, and $P_{[n,m-n]}$ with Levi subgroup $L_{[n,m-n]}\cong
GL(n)\times GL(m-n)$. For the parabolic subgroup $P_{[n,1^{m-n}]}$, the
generalized Springer fiber above $x\in\mathcal{O}_{\gamma}$, denoted by
$\mathcal{P}_{[n,1^{m-n}]}(x)$, is the set of $x$-stable partial flags
$V_{n}\subset\ldots\subset V_{m-1}\subset V_{m}$ in an $m$-dimensional vector
space $V_{m}$ whose lowest piece $V_{n}$ has dimension $n$, and for every two
consecutive spaces, we have $\mathrm{dim}(V_{i+1}/V_{i})=1$. The fiber
$\mathcal{P}_{[n,1^{m-n}]}(x)$ can be considered as

$\mathcal{P}_{[n,1^{m-n}]}(x)=\\{(x,gP_{[n,1^{m-n}]})\mid\mathrm{Ad}(g)^{-1}x\in\mathfrak{p}_{[n,1^{m-n}]}\\}.$

Similarly, for the parabolic subgroup $P_{[n,m-n]}$, the generalized Springer
fiber above $x\in\mathcal{O}_{\gamma}$, denoted similarly by
$\mathcal{P}_{[n,m-n]}(x)$, is a subvariety of $G/P_{[n,m-n]}$ given by

$\mathcal{P}_{[n,m-n]}(x)=\\{(x,gP_{n,m-n})\mid\mathrm{Ad}(g)^{-1}x\in\mathfrak{p}_{[n,m-n]}\\}.$

###  4.3. Springer Fibers, Jordan Forms, and the Proof of Part 2 of Theorem A

First, let’s consider the case of $\mathcal{P}_{[n,1^{m-n}]}(x)$ and
interprete this fiber as the collection of flags between $V_{n}$ and $V_{m}$.
Recalling the notations in Section 2.1, for any nilpotent element $x$, take
any skew Young tableau $T$ of the shape $\gamma/\alpha$ with the correponding
sequence of Young diagrams

$\alpha=\alpha_{0}\subset\alpha_{1}\subset\ldots\subset\alpha_{m-n}=\gamma$

such that
$\beta=(|\alpha_{1}/\alpha_{0}|,\ldots,|\alpha_{m-n}/\alpha_{m-n-1}|)$, we can
define a closed subvariety $\mathcal{P}^{T}(x)$ as the collections of
$x$-stable flags on which the nilpotent element $x$ acts on each stage
$V_{n+i}$ by a Jordan form $\lambda_{i}$, i.e. the restriction of $x$ on
$V_{n+i}$ has Jordan form given by a partition $\alpha_{i}$.

For any nilpotent element $x\in\mathcal{O}_{\gamma}$, there is a subvariety
$\mathcal{G}_{n}^{m}(x)$ of the Grassmannian variety $G(n,m)$ which
parametrizes the $x$-stable subspaces of $V_{m}$ of dimension $n$. By the
arguments in the Section 4 of [vL00], this subvariety of the Grassmannian
variety $G(n,m)$ can be decomposed into cells $\mathcal{G}_{\alpha,\beta}(x)$
such that the Jordan type of $x\mid_{V_{n}}$ on $V_{n}$ is $\alpha$, and the
Jordan type of $x^{\prime}$, the nilpotent linear operator acting on the
quotient space $V_{m}/V_{n}$ induced from $x$, is $\beta$:

$\mathcal{G}_{n}^{m}(x)=\bigcup_{\begin{subarray}{c}|\alpha|=n\\\
|\gamma|=m\\\
\alpha,\beta\subset\gamma\end{subarray}}\mathcal{G}_{\alpha,\beta}(x).$

By Lemma 4.1.1, each cell $\mathcal{G}_{\alpha,\beta}(x)$ is isomorphic to the
Satake fiber $\mathcal{S}^{\gamma}_{\alpha,\beta}$.
Moreover, the action of the nilpotent element $x$ with Jordan type $\beta$ on
the space $V_{m}/V_{n}$ corresponds to an $x$-stable filtration

$V_{n}\subset V_{n+1}\subset\ldots\subset V_{m}$

such that the action of $x$ on each quotient $V_{n+i}/V_{n}$ has Jordan form
$\alpha_{i}$. The collection of all the flags which satisfy this property can
be identified with the subspace $\mathcal{P}_{\alpha,\beta}(x)$ of the partial
flag variety $G/P_{[n,m-n]}$ defined below:

$\mathcal{P}_{\alpha,\beta}(x)=\left\\{gP\mid\mathrm{Ad}(g)^{-1}x\in\mathfrak{p}_{[n,m-n]},\text{
and
}p_{\mathfrak{l}}(\mathrm{Ad}(g)^{-1}x)\in\mathcal{O}_{\alpha,\beta}\right\\}$

where
$p_{\mathfrak{l}}:\mathfrak{p_{[n,m-n]}}\rightarrow\mathfrak{p_{[n,m-n]}}/\mathfrak{u}_{[n,m-n]}\cong\mathfrak{l}_{[n,m-n]}$
is the projection map. Letting $(\alpha,\beta)$ go over all the possible pairs
of Young diagrams satisfying $|\alpha|+|\beta|=|\gamma|$, the union of the
spaces $\mathcal{P}_{\alpha,\beta}(x)$ is exactly the generalized Springer
fiber $\mathcal{P}_{[n,m-n]}(x)$ over a point $x\in\mathcal{O}_{\gamma}$ of
the partial flag variety $G/P_{[n,m-n]}$. The correspondence between flags and
parabolic subgroups establishes isomorphisms between $\mathcal{G}_{n}^{m}(x)$
and $\mathcal{P}_{[n,m-n]}(x)$, and between $\mathcal{G}_{\alpha,\beta}(x)$
and $\mathcal{P}_{\alpha,\beta}(x)$.
Now we prove the second part of the Theorem 3.0.1:

###### Proof.

If $x\in\mathcal{O}_{\gamma}$ and
$gP_{[n,m-n]}\in\mathcal{P}_{\alpha,\beta}(x)$, each
$\mathrm{Ad}(g)^{-1}x\in\mathfrak{p}_{[n,m-n]}\cap\mathcal{O}_{\gamma}$, and
the space $\mathcal{P}_{\alpha,\beta}(x)$ is a closed subvariety of
$G/P_{[n,m-n]}$ which consists of those parabolic subalgebras
$\mathfrak{q}=\mathfrak{m}+\mathfrak{u}$ conjugate to $\mathfrak{p}_{[n,m-n]}$
such that

$x\in\mathfrak{q}\text{ and
}p_{\mathfrak{m}}\left(x\right)\in\mathcal{O}_{\alpha,\beta}.$

In [Spr78], [vL00] and the proof of the Lemma 3.4, it is stated that the
dimensions of the irreducible components of $\mathcal{P}_{\alpha,\beta}(x)$,
$\mathcal{G}_{\alpha,\beta}(x)$ and the Satake fiber
$S^{\gamma}_{\alpha,\beta}$ are equal to
$\langle\rho,\alpha+\beta-\gamma\rangle$, and the number of their components
is equal to the Littlewood-Richardson coefficient $c^{\gamma}_{\alpha,\beta}$.
Now we consider the subset $\mathcal{Q}_{\alpha,\beta}^{\gamma}$ of
$G/P_{[n,m-n]}$ consisting of parabolic subalgebras
$\mathfrak{q}=\mathfrak{m}+\mathfrak{u}$ such that

$\mathcal{O}_{\alpha,\beta}\subset
p_{\mathfrak{m}}(\mathcal{O_{\gamma}}\cap\mathfrak{q}),$

the fiber $\mathcal{P}_{\alpha,\beta}(x)$ is the intersection of
$\mathcal{Q}^{\gamma}_{\alpha,\beta}$ with the generalized Springer fiber
$\mathcal{P}_{[n,m-n]}(x)$:

$\mathcal{P}_{\alpha,\beta}(x)=\mathcal{Q}^{\gamma}_{\alpha,\beta}\cap\mathcal{P}_{[n,m-n]}(x).$

There is map

$\displaystyle\pi_{\alpha,\beta}^{\gamma}:G\supset\mathcal{P}_{\alpha,\beta}(x)\cdot
P$
$\displaystyle\longrightarrow\mathcal{O}_{\gamma}\cap\mathfrak{p}_{[n,m-n]}$
$\displaystyle g$ $\displaystyle\longmapsto\mathrm{Ad}(g)^{-1}x,$

which is the quotient by the centralizer $C_{G}(x)$ of $x$ from the left. We
denote the image of this map $\pi_{\alpha,\beta}^{\gamma}$ by
$\mathcal{P}_{\alpha,\beta}^{\gamma}$. In the $GL(n)$ case, the centralizer
$C_{G}(x)$ has only one connected component with dimension
$\sum_{i}(\gamma^{t})_{i}^{2}$. The number of connected components of
$\mathcal{P}_{\alpha,\beta}(x)\cdot P$ is still $c_{\alpha,\beta}^{\gamma}$,
since multiplying by an element $p\in P$ simply fixes a basis of the flag. By
(3) stated in Remark 3.2.3, the dimension of the connected components of
$\mathcal{P}_{\alpha,\beta}(x)\cdot P$ is equal to

$\dim\mathcal{P}_{\alpha,\beta}(x)\cdot
P=\frac{1}{2}\left(\sum_{i}(\gamma^{t})_{i}^{2}-\sum_{i}(\alpha^{t})_{i}^{2}-\sum_{i}(\beta^{t})_{i}^{2}\right)+\left(\sum\alpha_{i}^{t}\right)^{2}+\left(\sum\beta_{i}^{t}\right)^{2}+\left(\sum\alpha_{i}^{t}\right)\left(\sum\beta_{i}^{t}\right).$

Therefore, the dimension of $\mathcal{P}_{\alpha,\beta}^{\gamma}$ satisfies

$\dim\mathcal{P}_{\alpha,\beta}^{\gamma}=\dim\mathcal{P}_{\alpha,\beta}(x)\cdot
P-\dim C_{G}(x)>\dim\mathfrak{n}-\dim\mathcal{P}_{\alpha,\beta}(x)>0.$

The union of the closed orbits of $C_{G}(x)$ will not be Zariski-dense.
Therefore, the number of irreducible components of
$\mathcal{P}_{\alpha,\beta}^{\gamma}$ is also $c_{\alpha,\beta}^{\gamma}$.
Since we can interprete the image of the map as the elements in
$\mathcal{O}_{\gamma}\cap\mathfrak{p_{[n,m-n]}}$ whose projections along
$p_{\mathfrak{l}}$ lies in $\mathcal{O}_{\alpha,\beta}$, we have compeleted
the proof of the second part of the Theorem A. ∎

##  5\. Applications: Wave Front Sets and Associated Varieties

In this section, let $G$ be a semisimple algebraic group over an archimedean
field $F$. We would like to introduce invariants to describe of the “growth”
of irreducible representations $\pi$ of the group $G$. We denote the Lie
algebra of $G$ by $\mathfrak{g}_{0}$, and its complexified Lie algebra by
$\mathfrak{g}$.

###  5.1. Associated Varieties and Annihilator Varieties

For any irreducible representations $\pi$ of $G$, the “growth” of $\pi$ is
recorded in the _associated variety_ $\mathrm{As}(\pi)$ and its _annihilator
variety_ $\mathrm{An}(\pi)$. The algebra $U(\mathfrak{g})$ and its ideals are
filtered by the degrees, and the _annihilator variety_ $\mathrm{An}(M)$ and
the _associated variety_ $\mathrm{As}(M)$ of a module $M$ are defined as the
zero sets of the ideal $\mathrm{Gr}\left(\mathrm{Ann}(M)\right)$ and
$\mathrm{Ann}\left(\mathrm{Gr}(M)\right)$, respectively. For groups over
$\mathbb{R}$ and $\mathbb{C}$, it turns out that the annihilator variety
$\mathrm{An}(\pi)$ of any irreducible representation $\pi$ is the closure of
an unique orbit $\mathcal{O}(\pi)$ (cf. [Jos85, 3.10]).

###  5.2. Induction, Jacquet Functors and Associated Varieties

In [GS20], Gourevitch and Sayag discussed a family of Lagrangian submanifolds
of the annihilator variaties of any irreducible $U(\mathfrak{g})$-module.
According to [GS20, Corollary 6.6]:

######  Theorem 5.2.1.

For any irreducible representation $\pi$ of $G$ and a parabolic subgroup
$P=LN$, denoting by $r_{P}$ the Jacquet functor along the nilpotent radical
$N$, then for any irreducible quotient $\tau$ of $r_{P}(\pi)$,

$\mathcal{O}(\tau)\subset
p_{\mathfrak{l}}(\mathcal{O}(\pi)\cap\mathfrak{n}^{\perp}).$

Under the duality given by the Killing form, $\mathfrak{g}^{*}$ and
$\mathfrak{g}$ are isomorphic, and the following spaces correspond under such
isomorphism:

$\mathfrak{g}$ | $\mathfrak{g}^{*}\cong\mathfrak{g}$
---|---
$\mathfrak{n}^{\perp}$ | $\mathfrak{p}$
projection $p_{\mathfrak{l}}:\mathfrak{p}\rightarrow\mathfrak{p}/\mathfrak{n}$ | projection $q_{\mathfrak{l}}:\mathfrak{n}^{\perp}\rightarrow\mathfrak{n}^{\perp}/\mathfrak{p}^{\perp}$
$\mathcal{O}(\pi)\cap\mathfrak{n}^{\perp}$ | $\mathcal{O}(\pi)\cap\mathfrak{p}$
$\mathfrak{n}^{\perp}\cap p_{\mathfrak{l}}^{-1}\mathcal{O}(\tau)$ |  $\mathcal{O}(\tau)+\mathfrak{n}$.

Therefore, under the Killing form duality, the following two inclusions are
equivalent:

$\mathcal{O}(\tau)\subset
p_{\mathfrak{l}}\left(\mathcal{O}(\pi)\cap\mathfrak{n}^{\perp}\right)\leftrightharpoons\mathcal{O}(\pi)\subset
G\cdot(\mathcal{O}(\tau)+\mathfrak{n}).$

###  5.3. Examples

In this section we will give some examples and applications to the Theorem A.

####  5.3.1. $n=3$

Consider the standard parabolic subgroup $P_{[21]}=L_{[21]}N_{[21]}$ where the
Levi subgroup $L_{[21]}=GL(2)\times GL(1)$, and the unipotent radical

$N_{[21]}=\begin{pmatrix}1&0&*\\\ 0&1&*\\\ 0&0&1\end{pmatrix}.$

The generalized flag variety $G/P_{[21]}$ is isomorphic to $\mathbb{P}^{2}$,
and the isomorphism can be written explicitly down with the Plücker
coordinates

$\displaystyle G/P_{[21]}$
$\displaystyle\longrightarrow\mathbb{P}\left({\wedge^{2}\mathbb{C}^{3}}\right)$
$\displaystyle[g]$ $\displaystyle\mapsto(g(e_{1}\wedge e_{2}),g(e_{2}\wedge
e_{3}),g(e_{3}\wedge e_{1})).$

For an element $x=\left(\begin{smallmatrix}0&0&0\\\ 0&0&1\\\
0&0&0\end{smallmatrix}\right)\in\mathcal{O}_{[21]}$, the generalized Springer
fiber $\mathcal{P}_{[21]}(x)$ above $x$ is isomorphic to the projective line
$\mathbb{P}^{1}$, and is given by the equation $X_{1}=0$ in the Plücker
coordinate $[X_{0}:X_{1}:X_{2}]$. The generalized Springer fiber above an
element $\left(\begin{smallmatrix}0&1&0\\\ 0&0&1\\\
0&0&0\end{smallmatrix}\right)$ is the point $[1:0:0]$.
For the case $\alpha=[11]$ and $\beta=[1]$, the two possible choices of
$\gamma$ are $[21]$ and $[111]$ with Littlewood-Richardson coefficients
$c_{[11],[1]^{[21]}}=c_{[11],[1]^{[111]}}=1$. The orbit $\mathcal{O}_{[111]}$
is the zero orbit, hence for all parabolic subalgebras
$\mathfrak{q}=\mathfrak{m}+\mathfrak{u}$ conjugate to $\mathfrak{p}_{[21]}$,
the variety $p_{\mathfrak{m}}\left(\mathcal{O}_{[111]}\cap\mathfrak{q}\right)$
contains the zero orbit $\mathcal{O}_{[11]}\times\mathcal{O}_{[1]}$.

The orbit $\mathcal{O}_{[21]}$ can be explicitly described with matrices:

$\mathcal{O}_{[21]}=\left\\{A\in\mathfrak{g}\mid A^{2}=0\text{ and }A\neq
0\right\\}.$

We can represent any element in $\mathfrak{p}_{[21]}$ as a matrix of the shape
$\begin{pmatrix}A&v\\\ 0&b\end{pmatrix}$ where $v\in\mathbb{C}^{2}$. The
subvariety $\mathcal{O}_{[21]}\cap\mathfrak{p}_{[21]}$ is the following subset

$\mathcal{O}_{[21]}\cap\mathfrak{p}_{[21]}=\left\\{\begin{pmatrix}A&v\\\
0&0\end{pmatrix}\neq 0\mid A^{2}=0,Av=0\right\\}$

of $\mathfrak{p}_{[21]}$. The elements $[g]$ in the generalized Springer fiber
$\mathcal{P}_{[21]}(x)$ are those who satisfy
$\mathrm{Ad}(g)^{-1}x\in\mathcal{O}_{[21]}\cap\mathfrak{p}_{[21]}$. The point
$[0:0:1]\in\mathcal{P}_{[21]}(x)$ projects to the zero orbit
$\mathcal{O}_{[11]}$ and the affine chart $[1:0:x]$ projects to
$\mathcal{O}_{[2]}$.

The orbit $\mathcal{O}_{[3]}$ is the set of matrices satisfying

$\mathcal{O}_{[3]}=\left\\{A\in\mathfrak{g}\mid A^{3}=0,\;A^{2}\neq 0\text{
and }A\neq 0\right\\}.$

Therefore, the subvariety $\mathcal{O}_{[3]}\cap\mathfrak{p}_{[21]}$ is the
following set of matrices:

$\mathcal{O}_{[3]}\cap\mathfrak{p}_{[21]}=\left\\{\begin{pmatrix}A&v\\\
0&0\end{pmatrix}\neq 0\mid A^{3}=0,A^{2}v=0,\;A^{2}\neq 0\text{ or }Ab\neq
0\right\\}.$

The generalized Springer fiber over a point of $\mathcal{O}_{[3]}$ is the
point $[1:0:0]$, and the whole set $\mathcal{O}_{[3]}\cap\mathfrak{p}_{[21]}$
projects to the orbit $\mathcal{O}_{[2]}$.

####  5.3.2. $n=6$

Consider the Levi blocks $GL(4)\times GL(2)\subset GL(6)$, and consider the
orbit $\mathcal{O}_{[22]}\times\mathcal{O}_{[11]}$. Between the induced orbit
$[33]$ and $[2211]$, the poset of $G$-orbits contained in
$G\cdot(\mathcal{O}_{[22],[11]}+\mathfrak{n})$ is displayed as below:

$[33]$$[321]$$[3111]$$[222]$$[2211]$

The orbits $[3111]$ and $[222]$ are excluded from the poset, since
$c_{33,2211}^{3111}=c_{33,2211}^{222}=0$.
It is important to mention that the $n=6$ case for $P_{[33]}$ is the first
case in which we can expect a Littlewood-Richardson coefficient greater than
1. The partial flag variety $G/P_{[33]}$ is isomorphic to the Grassmannian
$G(3,6)$ and can be embedded into the projective space
$\mathbb{P}(\bigwedge^{3}\mathbb{C}^{6})\cong\mathbb{P}^{19}$ with the Plücker
embeddings by choosing the basis of $\bigwedge^{3}\mathbb{C}^{6}$ as
$e_{i}\wedge e_{j}\wedge e_{k}$ for each triple $i<j<k$.
We will discuss a slightly different space $\mathcal{P}_{\alpha,\beta}(x,y)$
by taking those elements $[g]$ in $\mathcal{P}_{\alpha,\beta}(x)$ such that
$p_{\mathfrak{l}}(\mathrm{Ad}^{-1}(g)x)$ is a fixed element
$y\in\mathcal{O}_{\alpha}\times\mathcal{O}_{\beta}$. There is a map

$\sigma:\mathcal{P}_{\alpha,\beta}(x,y)\cdot
P\rightarrow\mathrm{Spec}(\mathbb{C}[\mathfrak{p}]^{L})$

which takes every $g$ to $\mathrm{Ad}(g)^{-1}x$, and the ring
$\mathbb{C}[\mathfrak{p}]^{L}$ is the ring of $\mathrm{Ad}L$-invariant regular
functions on $\mathfrak{p}$ under the adjoint action of $L$. Any element of
$\mathrm{Spec}(\mathbb{C}[\mathfrak{p}]^{L})$ lying in the image of $\sigma$
can be represented by a matrix

$\left(\begin{array}[]{cccccc}0&1&0&u_{1,4}&u_{1,5}&u_{1,6}\\\
0&0&0&u_{2,4}&u_{2,5}&u_{2,6}\\\ 0&0&0&u_{3,4}&u_{3,5}&u_{3,6}\\\
0&0&0&0&1&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ \end{array}\right).$

The images of the map $\sigma$ for different choices of $\gamma$ are listed in
the following table:

$\gamma$ | $\dim\mathcal{P}_{\alpha,\beta}(x)$ | $c_{[21],[21]}^{\gamma}$ | Ideal
---|---|---|---
$[2211]$ | 5 | 1 | $(u_{3,6},u_{2,6},u_{2,4},u_{3,4},u_{1,4}+u_{2,5})$
$[222]$ | 4 | 1 | $(u_{1,4}+u_{3,5},u_{2,4},u_{3,4},u_{2,6})$
$[3111]$ | 4 | 1 | $(u_{3,6},u_{2,6},u_{2,4},u_{3,4})$
$[3,2,1]$ | 2 | 2 | $(u_{3,4},u_{2,4}),(u_{2,6},u_{2,4})$
$[3,3]$ | 1 | 1 | $(u_{2,4})$
$[4,1,1]$ | 1 | 1 | $(u_{2,6}u_{3,4}-u_{2,4}u_{3,6})$
$[4,2]$ | 0 | 1 | $(0)$

.

####  5.3.3. $n=8$

We look at the Levi blocks $GL(4)\times GL(4)\subset GL(8)$, and consider the
orbit $\mathcal{O}_{[22]}\times\mathcal{O}_{[22]}$. Between the induced orbit
$[44]$ and $[2222]$, the poset of $G$-orbit
$G\cdot(\mathcal{O}_{[22],[22]}+\mathfrak{n})$ is displayed as below:

$[44]$$[431]$$[422]$$[4211]$$[332]$$[3311]$$[3221]$$[2222]$

The orbits $[4211]$ and $[332]$ are excluded from the poset, since
$c_{22,22}^{4211}=c_{22,22}^{332}=0$.

####  5.3.4. $n=12$

Consider the Levi blocks $GL(6)\times GL(6)\subset GL(12)$ and the orbit
$\mathcal{O}_{[321]}\times\mathcal{O}_{[321]}$, if we represent any element of
$\mathrm{Spec}(\mathbb{C}[\mathfrak{p}]^{L})$ by the matrix

$\left(\begin{array}[]{cccccccccccc}0&1&0&0&0&0&u_{1,7}&u_{1,8}&u_{1,9}&u_{1,10}&u_{1,11}&u_{1,12}\\\
0&0&1&0&0&0&u_{2,7}&u_{2,8}&u_{2,9}&u_{2,10}&u_{2,11}&u_{2,12}\\\
0&0&0&0&0&0&u_{3,7}&u_{3,8}&u_{3,9}&u_{3,10}&u_{3,11}&u_{3,12}\\\
0&0&0&0&1&0&u_{4,7}&u_{4,8}&u_{4,9}&u_{4,10}&u_{4,11}&u_{4,12}\\\
0&0&0&0&0&0&u_{5,7}&u_{5,8}&u_{5,9}&u_{5,10}&u_{5,11}&u_{5,12}\\\
0&0&0&0&0&0&u_{6,7}&u_{6,8}&u_{6,9}&u_{6,10}&u_{6,11}&u_{6,12}\\\
0&0&0&0&0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&1&0\\\
0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0\\\ \end{array}\right)$

If we set $\gamma=[53211]$, the ideals for the image of the projection map
from the 6 dimensional Satake fiber with 4 irreducible components are the
following four ideals:

$\displaystyle I_{1}$
$\displaystyle=(u_{6,10},u_{6,7},u_{5,10},u_{5,7},u_{3,10},u_{3,7})$
$\displaystyle I_{2}$ $\displaystyle=(u_{6,7},u_{5,7},u_{3,7},$ $\displaystyle
u_{5,12}u_{6,10}-u_{5,10}u_{6,12},u_{3,12}u_{6,10}-u_{3,10}u_{6,12},u_{3,12}u_{5,10}-u_{3,10}u_{5,12}$
$\displaystyle
u_{3,12}u_{4,7}+u_{3,12}u_{5,8}-u_{2,7}u_{5,12}-u_{3,8}u_{5,12},$
$\displaystyle
u_{3,10}u_{4,7}+u_{3,10}u_{5,8}-u_{2,7}u_{5,10}-u_{3,8}u_{5,10})$
$\displaystyle I_{3}$
$\displaystyle=(u_{5,12},u_{5,10},u_{5,7},u_{3,12},u_{3,10},u_{3,7})$
$\displaystyle I_{4}$ $\displaystyle=(u_{3,12},u_{3,10},u_{3,7},$
$\displaystyle
u_{5,12}u_{6,10}-u_{5,10}u_{6,12},u_{5,12}u_{6,7}-u_{5,7}u_{6,12},u_{5,10}u_{6,7}-u_{5,7}u_{6,10}$
$\displaystyle
u_{2,10}u_{6,7}+u_{3,11}u_{6,7}-u_{2,7}u_{6,10}-u_{3,8}u_{6,10},$
$\displaystyle
u_{2,10}u_{5,7}+u_{3,11}u_{5,7}-u_{2,7}u_{5,10}-u_{3,8}u_{5,10}).$

The Littlewood-Richardson coefficient $c_{[321],[321]}^{[53211]}$ of this case
is equal to 4.

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