# A categorification of the Lusztig–Vogan module

Scott Larson and Anna Romanov

###### Abstract.

We construct two categorifications of the Lusztig–Vogan module associated to a
real reductive algebraic group. The first categorification is given by
semisimple complexes in an equivariant derived category, and the second is
constructed as a module category over Soergel bimodules. Our categorifications
are related by taking equivariant hypercohomology.

###### Contents

  1. 1 Introduction
    1. 1.1 Overview
    2. 1.2 Geometry of the Lusztig–Vogan module
    3. 1.3 A Soergel bimodule approach
    4. 1.4 Application: Resolutions of singularities
    5. 1.5 Relation to the results of Bezrukavnikov–Vilonen
    6. 1.6 Acknowledgements
  2. 2 Index of notation
  3. 3 Preliminaries
    1. 3.1 The Hecke algebra
    2. 3.2 The Lusztig–Vogan module
    3. 3.3 Soergel bimodules
    4. 3.4 The equivariant derived category
    5. 3.5 The geometric Hecke category
    6. 3.6 From the geometric Hecke category to Soergel bimodules
  4. 4 A geometric categorification
    1. 4.1 Semisimple complexes
    2. 4.2 Local systems
    3. 4.3 Proof of Theorem 4.1.3
    4. 4.4 Application: Resolutions of singularities
  5. 5 A categorification of the trivial block
    1. 5.1 Blocks
    2. 5.2 The category generated by closed orbits
  6. 6 An algebraic categorification
    1. 6.1 Torus fixed points and closed $K$-orbits
    2. 6.2 A category of bimodules
  7. 7 The hypercohomology functor
    1. 7.1 The essential image of the hypercohomology functor
    2. 7.2 Hypercohomology is fully faithful
  8. A Dual pairs in monoidal categories

##  1\. Introduction

###  1.1. Overview

Let $G$ be a connected complex reductive algebraic group, $\theta$ a
holomorphic involution of $G$, and $K$ a finite-index subgroup of the fixed
point group $G^{\theta}$. From this data, Lusztig–Vogan constructed a module
over the Hecke algebra of $G$ and a collection of polynomials that arise as
entries in a change-of-basis matrix for this module [Vog82, LV83]. These
Kazhdan–Lusztig–Vogan (KLV) polynomials have both geometric and representation
theoretic significance: geometrically, they capture subtle information about
the singularity of closures of $K$-orbits in the flag variety of $G$, and
representation theoretically, they give character formulas for irreducible
representations of the associated real reductive group $G_{\mathbb{R}}$ (cf.,
e.g., [AvLTV20]).

In this paper, we take a Soergel bimodule approach to the Lusztig–Vogan
module. In classic work [Soe90, Soe00], Soergel explained that the
hypercohomology functor applied to certain categories of constructible sheaves
appearing in Lie theory is fully faithful. The upshot is that one can
understand categories of constructible sheaves algebraically, as certain
modules or bimodules. In this paper, we explain that a piece of the
Lusztig–Vogan module can also be understood in this way after taking
hypercohomology of a category of constructible sheaves.

A major motivation for Soergel’s work was to establish certain Koszul duality
equivalences on the derived category of category $\mathcal{O}$. Similarly, one
motivation for giving a bimodule theoretic interpretation of the Lusztig–Vogan
module might be to establish Soergel’s conjectures [Soe01], which generalize
Koszul duality to real reductive groups. It is this point of view that has
been taken in recent work of Bezrukavnikov–Vilonen [BV21], which has results
close to those of this paper111We comment on these results in more detail in
Section 1.5.

Another motivation for our work is to understand as explicitly as possible the
Lusztig–Vogan module from a Soergel bimodule point of view, as a rich new
source of module categories for Soergel bimodules.

###  1.2. Geometry of the Lusztig–Vogan module

The remainder of the introduction describes our results in more detail. Fix a
Borel subgroup $B$ of $G$. The group $K$ acts on the flag variety $X=G/B$ with
finitely many orbits, and the Lusztig–Vogan module is constructed using the
geometry of these orbits. Denote by $\mathscr{D}$ the set of pairs
$(\mathcal{O},\mathcal{L})$ consisting of a $K$-orbit $\mathcal{O}$ and an
irreducible $K$-equivariant local system on $\mathcal{O}$, and set $M_{LV}$ to
be the free $\mathbb{Z}[q^{\pm 1}]$-module with basis $\mathscr{D}$. Let
$(W,S)$ be the Coxeter system associated to $(G,B)$ and $\bm{H}=\bm{H}(W,S)$
the corresponding Hecke algebra. For each $s\in S$, one can define
endomorphisms

$T_{s}:M_{LV}\rightarrow M_{LV}$

using push-pull functors to partial flag varieties or via a case-by-case
analysis of root types (Definition 3.2.1). The operators $T_{s}$ give $M_{LV}$
the structure of an $\bm{H}$-module.

The geometric construction of $M_{LV}$ leads naturally to a categorification
within $D^{b}_{K}(X)$, the $K$-equivariant derived category of sheaves of
$\mathbb{Q}$-vector spaces222For simplicity, we use $\mathbb{Q}$-coefficients
throughout; however, our arguments would also work for any field of
characteristic zero. on $X$. Associated to each Lusztig–Vogan parameter
$(\mathcal{O},\mathcal{L})\in\mathscr{D}$ is an indecomposible object in
$D^{b}_{K}(X)$: the $K$-equivariant intersection cohomology complex of
$\overline{\mathcal{O}}$ with coefficients in $\mathcal{L}$ (shifted by
$\dim\mathcal{O}$) which we denote by
$\bm{IC}(\overline{\mathcal{O}},\mathcal{L})$. We define

$\mathcal{M}_{LV}^{ss}=\langle\bm{IC}(\overline{\mathcal{O}},\mathcal{L})\mid(\mathcal{O},\mathcal{L})\in\mathscr{D}\rangle_{\oplus,[1]}$

to be the additive subcategory of $D^{b}_{K}(X)$ generated by $\bm{IC}$
sheaves under direct sum ($\oplus$) and grading shift ($[1]$). Via right
convolution, $D^{b}_{K}(X)$ admits a categorical action of the $B$-equivariant
derived category $D^{b}_{B}(X)$. Thanks to the decomposition theorem,
$\mathcal{M}_{LV}^{ss}$ is preserved under this action, and thus obtains a
categorical action of the geometric Hecke category $\mathcal{H}$ (Definition
3.5.1). Passing to Grothendieck groups333The notation $[\mathcal{A}]_{\oplus}$
denotes the split Grothendieck group of an additive category $\mathcal{A}$, we
recover the Lusztig–Vogan module.

######  Theorem 1.2.1.

(Theorem 4.1.3) As right $\bm{H}$-modules,
$[\mathcal{M}_{LV}^{ss}]_{\oplus}\simeq M_{LV}$.

Recall that, by the Beilinson–Bernstein localization theorem, modules for the
Lie algebra $\mathfrak{g}$ of $G$ may be understood as global sections of
$D$-modules on the flag variety. The irreducible $(\mathfrak{g},K)$-modules
with trivial infinitesimal character occur as the global sections of
$D$-modules corresponding to $\bm{IC}(\overline{\mathcal{O}},\mathcal{L})$
under the Riemann–Hilbert correspondence. In this way, the set $\mathscr{D}$
parametrizes certain equivalence classes of irreducible representations of the
real reductive group corresponding to the Harish-Chandra pair
$(\mathfrak{g},K)$. This was a major motivation for the work of Lusztig and
Vogan, and leads to the block decomposition (Definition 5.1.1, Remark 5.1.2)
of the Lustig–Vogan module. Blocks of $M_{LV}$ keep track of which irreducible
$(\mathfrak{g},K)$-modules are related via composition series of standard
modules. The block of the trivial representation, $M_{LV}^{0}$, arises in the
geometric formulation above as the submodule of $M_{LV}$ generated by trivial
local systems on closed orbits (Theorem 5.1.3). Set

$\mathcal{M}_{LV}^{0}:=\langle\bm{IC}(\mathcal{Q},\mathbb{Q}_{\mathcal{Q}})*\mathcal{H}\mid\mathcal{Q}\text{
a closed $K$-orbit}\rangle_{\oplus,\ominus,[1]}$

to be the subcategory of $\mathcal{M}_{LV}^{ss}$ generated by $\bm{IC}$
sheaves corresponding to trivial local systems on closed orbits. (Here
$\mathbb{Q}_{\mathcal{Q}}$ is the constant sheaf on $\mathcal{Q}$ and the
subscript $\oplus,\ominus,[1]$ denotes the the Karoubi envelope.)

######  Theorem 1.2.2.

(Theorem 3.5.2) As right $\bm{H}$-modules,
$[\mathcal{M}_{LV}^{0}]_{\oplus}\simeq M_{LV}^{0}.$

The categories $\mathcal{M}_{LV}^{ss}$ and $\mathcal{M}_{LV}^{0}$ provide
geometric categorifications of the Lusztig–Vogan module $M_{LV}$ and its
trivial block $M_{LV}^{0}$. The existence of these categorifications is not
surprising—they are just reformulations of the construction in [LV83] and
classical theorems in [Vog81] using different geometric language. Our interest
in formulating the Lusztig–Vogan module in these geometric terms is twofold:
(1) we believe that the equivariant derived category provides the most natural
language to describe Lusztig–Vogan’s construction, and (2) this formulation
allows us to give a bimodule description of the Lusztig–Vogan module which
relates it to the monoidal category of Soergel bimodules.

###  1.3. A Soergel bimodule approach

Our tool for moving from the $K$-equivariant derived category to a category of
bimodules is the hypercohomology functor (Definition 3.16):

$\mathbb{H}_{K}^{\bullet}(-):=\operatorname{Hom}_{K}^{\bullet}(\mathbb{Q}_{X},-):D^{b}_{K}(X)\rightarrow
H^{*}_{K}(X)\mathrm{-gmod}.$

To make this approach work, we add the assumption $K$ is the identity
component of $G^{\theta}$. We suspect that the main results hold for
disconnected $K$ as well, but different arguments would be needed. Under this
connectedness assumption on $K$, a standard computation (Lemma 7.0.1) proves
that the $K$-equivariant cohomology of the flag variety is

$H^{*}_{K}(X)=P^{W_{K}}\otimes_{R^{W}}R,$

where $P=S(X(T_{K})_{\mathbb{Q}})$ is the symmetric algebra on the
$\mathbb{Q}$-span of the cocharacter lattice of a maximal torus $T_{K}$ in
$K$, $R=S(X(T)_{\mathbb{Q}})$ is the analogous ring for a maximal
$\theta$-stable torus $T$ in $G$, and $W_{K}$ is the Weyl group of $K$. This
allows us to view vector spaces in the image of $\mathbb{H}^{\bullet}_{K}$ as
graded $(P^{W_{K}},R)$-bimodules.

Our algebraic Lusztig–Vogan category is defined to be the essential image of
the functor $\mathbb{H}_{K}^{\bullet}$ when restricted to
$\mathcal{M}_{LV}^{0}$. It can be described in an explicit combinatorial way
by computing the hypercohomology of $\bm{IC}$ sheaves on closed orbits (Lemma
7.1.2). For a closed $K$-orbit $\mathcal{Q}$, there exists an element $x\in W$
such that

$\mathbb{H}_{K}^{\bullet}(\bm{IC}(\mathcal{Q},\mathbb{Q}_{\mathcal{Q}}))=P_{x},$

where $P_{x}$ is the standard $(P^{W_{K}},R)$-bimodule (Definition 6.2.1)
corresponding to $x$. We define a category

$\mathcal{N}_{LV}^{0}:=\langle P_{x}\otimes_{R}\mathbb{S}\text{Bim}\mid x\in
W^{\theta}\rangle_{\oplus,\ominus,(1)}$

of $(P^{W_{K}},R)$-bimodules which is generated by standard modules under the
natural right action of the category of Soergel bimodules (denoted
$\mathbb{S}\text{Bim}$, Definition 3.3.2). Here $W^{\theta}$ is the set of
elements of $W$ fixed under the involution induced from $\theta$. These are
precisely the elements of $W$ corresponding to $T$-fixed points in $X$ which
are found in closed $K$-orbits.

Though motivated by geometry, this construction can be done completely
combinatorially with only the input data of the group $G$ and involution
$\theta$. From the perspective of Soergel bimodules, this provides an
interesting class of module categories for $\mathbb{S}\text{Bim}$ coming from
real groups. It also potentially leads to “exotic” Lusztig–Vogan modules
attached to “exotic” involutions (e.g. of non-crystallographic Weyl groups).

The main result of our paper is the following theorem.

######  Theorem 1.3.1.

(Theorem 7.1.1, Theorem 7.2.1) The hypercohomology functor

$\mathbb{H}_{K}^{\bullet}:\mathcal{M}_{LV}^{0}\rightarrow\mathcal{N}_{LV}^{0}$

is an equivalence of categories.

######  Remark 1.3.2.

This result can also be deduced from Theorem 3.12 in [BV21].

The proof of Theorem 1.3.1 relies on two main ingredients: the theory of
parity sheaves developed in [JMW14], and localization to torus fixed points as
in [GKM98]. We find it interesting that this appears to be the first
application of parity sheaf techniques to $K$-orbits on flag varieties, as
they provide a powerful tool which has been used in many other settings in
geometric representation theory. It is also of note that when specialized to
the case of a complex group (i.e., the involution of $G\times G$ which
switches the factors), our argument provides an alternate proof of [Soe90,
Erweiterungssatz 17] which does not rely on known facts about dimensions of
modules in category $\mathcal{O}$.

Theorem 1.3.1 immediately implies that $\mathcal{N}_{LV}^{0}$ categorifies the
trivial block of the Lusztig–Vogan module.

######  Corollary 1.3.3.

(Corollary 7.2.17) As right $\bm{H}$-modules,
$[\mathcal{N}_{LV}^{0}]_{\oplus}\simeq M_{LV}^{0}$.

A conjecture of Soergel [Soe01] predicts that associated to the category
$\mathcal{N}_{LV}^{0}$ (or $\mathcal{M}_{LV}^{0}$) is a Koszul dual category
of Harish-Chandra modules for a real form of the Langlands dual group of $G$.
The duality on the level of categories should provide a categorification of
Vogan’s character duality [Vog83], which is a duality on the level of
$\bm{H}$-modules. A version of this relationship is established in [BV21].

One might hope that by applying the functor $\mathbb{H}_{K}^{\bullet}$ to the
semisimple category $\mathcal{M}_{LV}^{ss}$, we could establish a bimodule
description of the entire Lusztig–Vogan module. Unfortunately,
$\mathbb{H}_{K}^{\bullet}$ vanishes outside of $\mathcal{M}_{LV}^{0}$ (see
Example 7.1.6), so this strategy does not work.

###  1.4. Application: Resolutions of singularities

We can view the results in this paper as studying continuous actions of real
reductive algebraic groups. Gelfand’s program asserts that one should first
linearize such actions by considering vector spaces related to functions on
the original space. This philosophy lead to the definition of $\mathscr{D}$
through work of Harish-Chandra and Vogan [Vog82]. In particular, continuous
actions of $G_{\mathbb{R}}$ are studied by considering the easier problem of
an algebraic group $K$ acting on an algebraic variety $X$. It is interesting
to observe that in the case of $G_{\mathbb{R}}$ acting on the flag variety,
Matsuki duality gives a perfect relationship between orbit structures (cf.,
[Wol69],[Mat79]). Moreover, singularities of $K$-orbit closures in $X$ are
related to representation theory of $G_{\mathbb{R}}$ through KLV polynomials
and characteristic cycles of $D$-modules.

Closures of $K$-orbits in $X$ have complicated singularities. In [BE94],
equivariant resolutions of singularities were constructed to study the
geometry of these varieties, and in [Lar20] equivariant proper maps were
further analyzed. As an application of our geometric categorification of the
Lusztig–Vogan module, we give a formula in Theorem 4.4.1 to compute the
cohomology of fibers of the resolutions introduced in [Lar20] using the
character map

(1.1) $m\,\colon D_{K}^{b}(X)\to M_{LV}$

appearing in the proof of Theorem 4.1.3. This result can be viewed through the
paradigm of Gelfand and Harish-Chandra as studying equivariant resolutions of
singularities first via linearization, then through algebraic methods using
our algebraic categorification.

###  1.5. Relation to the results of Bezrukavnikov–Vilonen

Recent results of Bezrukavnikov–Vilonen [BV21] establish an important special
case of Soergel’s conjecture on a categorification of Vogan duality (see 1.1).
One side of their equivalence takes place in the equivariant derived category,
and a key part of their argument involves the fully-faithfullness of a global
sections functor. This part of their argument involves a theorem equivalent to
Theorem 1.3.1 above. Our work on this project began before we were aware of
their results, and our proofs differ. (In particular, their proof involves a
reduction to low ranks, and ours does not, see 1.5.1 below). For the benefit
of the reader, we comment here on the relation between our results.

The fully-faithfullness result alluded to above is [BV21, Thm. 3.2]. They
prove that (in their notation)

$R\Gamma_{\check{K}}:\mathcal{S}^{\mathrm{gr}}\rightarrow\mathrm{Coh}^{\mathbb{C}^{\times}}(\mathfrak{B})$

is fully faithful. The relation between this result and Theorem 1.3.1 is as
follows.

  1. (1)

Their category $\mathcal{S}^{\mathrm{gr}}$ is equivalent to our category
$\mathcal{M}_{\mathrm{LV}}^{0}$. Indeed, both categories consist of certain
$K$-equivariant semisimple complexes on $G/B$, and [BV21, Prop. 3.6]
establishes that $\mathcal{S}^{\mathrm{gr}}$ is generated by $\bm{IC}$-sheaves
corresponding to trivial local systems on closed orbits.

  2. (2)

There is an equivalence

(1.2) $\mathrm{Coh}^{\mathbb{C}^{\times}}(\mathfrak{B})\simeq
P^{K}\otimes_{R^{W}}R\mathrm{-gmod},$

where we are using notation from [BV21] on the left and our notation on the
right. This is because $\mathbb{C}[\mathfrak{B}]=P^{K}\otimes_{R^{W}}R$ by
[BV21, Lem. 3.1]. The grading on the right hand side of (1.2) is due to the
$\mathbb{C}^{\times}$-action on the left hand side of (1.2). The weights agree
by the definition of the $\mathbb{C}^{\times}$-action in [BV21, Thm. 3.2].

  3. (3)

In [BV21], the category $\mathcal{S}^{\mathrm{gr}}$ is defined using
$K$-equivariant $D$-modules on $G/B$. It is then implicitly identified with a
subcategory of the (constructible) equivariant derived category
$D^{b}_{K}(G/B)$. In contrast, we work exclusively within the equivariant
derived category.

We complete this section with some further remarks.

######  Remark 1.5.1.

  1. (i)

Bezrukavnikov–Vilonen’s proof uses reduction to small rank and a computation,
see [BV21, §3.8]. Our proof avoids this reduction.

  2. (ii)

Our theorem describes the image in terms of Soergel bimodules. One can deduce
a similar description from [BV21]. Note that our Soergel bimodule action
corresponds to their convolution with $\alpha$-lines, see [BV21, §3.3].

  3. (iii)

For the purposes of Bezrukavnikov–Vilonen, it is necessary to consider all
finite-index subgroups of $K$, included disconnected ones. They overcome this
obstacle by imposing equivariance under a finite abelian group.

###  1.6. Acknowledgements

We would like to thank Jeff Adams, Adam Brown, Bill Graham, Dragan Milicic,
Laura Rider, Peter Trapa, David Vogan and Geordie Williamson for useful
discussions and comments. We would like to thank the University of Georgia for
providing us the opportunity to first meet and discuss $K$-orbits, and the
first author further thanks this university for providing an environment
conducive to learning material related to this paper. The second author
completed the final stages of this work on the University of Sydney’s One Tree
Island research station, which she would like to thank for providing an
exceptional working environment. The second author acknowledges support from
the National Science Foundation Award No. 1803059 and the Australian Research
Council grant DP170101579.

##  2\. Index of notation

$G$: connected reductive complex algebraic group (1.1)

$\theta$: holomorphic involution of $G$ (1.1)

$K$: finite-index subgroup of $G^{\theta}$ in Sections 3 \- 5, the identity
component of $G^{\theta}$ in Sections 6 \- 7

$T_{K}$: maximal torus in $K$ (6.1)

$T=Z_{G}(T_{K})$: $\theta$-stable maximal torus in $G$ (Lemma 6.1.2)

$B$: Borel subgroup of $G$ containing $T$ (1)

$B_{K}=B\cap K$: Borel subgroup of $K$ (6.1)

$\mathfrak{g},\mathfrak{k},\mathfrak{t},\mathfrak{t}_{K}$: Lie algebras of
$G,K,T,T_{K}$ (6.1)

$W=N_{G}(T)/T$, $W_{K}=N_{K}(T_{K})/T_{K}$: Weyl groups of $G,K$ (Lemma 6.1.3)

$W^{\theta}$: group of elements fixed by the involution on $W$ induced by
$\theta$ (6.1)

$\bm{H}=\bm{H}(W,S)$: Hecke algebra of $(W,S)$ (3.1)

$\\{\delta_{x}\mid x\in W\\}$: standard basis of the Hecke algebra $\bm{H}$
(3.1)

$\\{b_{x}\mid x\in W\\}$: Kazhdan–Lusztig basis of the Hecke algebra $\bm{H}$
(Theorem 3.1.2)

$X(T),X(T_{K})$: character lattices of $T$, $T_{K}$ (1.3)

$X(T)_{\mathbb{Q}}=X(T)\otimes_{\mathbb{Z}}\mathbb{Q}$,
$X(T_{K})_{\mathbb{Q}}=X(T_{K})\otimes_{\mathbb{Z}}\mathbb{Q}$:
$\mathbb{Q}$-vector spaces spanned by character lattices (1.3)

$R=S(X(T)_{\mathbb{Q}})$: Symmetric algebra on $X(T)_{\mathbb{Q}}$ (3.3)

$P=S(X(T_{K})_{\mathbb{Q}})$: Symmetric algebra on $X(T_{K})_{\mathbb{Q}}$ (6)

$B_{s}=R\otimes_{R^{s}}R(1)$,
$BS(\underline{w})=B_{s_{1}}\otimes_{R}\cdots\otimes_{R}B_{s_{n}}$:
Bott–Samelson bimodules corresponding to $s\in S$ and
$\underline{w}=(s_{1},\ldots s_{n})$ (Equation 3.11, Definition 3.3.1)

$\mathbb{S}\text{Bim}$: category of Soergel bimodules (Definition 3.3.2)

$P^{K}:=P^{W_{K}}$: $W_{K}$-invariant polynomials in $P$ (6)

$P_{x}$: standard bimodule in $(P^{K},R)\mathrm{-gbim}$ corresponding to $x\in
W$ (Definition 6.2.1)

$\mathcal{N}_{LV}^{0}$: subcategory in $(P^{K},R)\mathrm{-gbim}$ generated by
standard bimodules (Definition 6.2.5)

$X=G/B$: the flag variety of $G$ (3.5)

$X_{w}=BwB/B$: Schubert cell corresponding to $w\in W$

$\mathcal{H}$: the geometric Hecke category (Definition 3.5.1, Equation 3.21)

$\bm{IC}_{w}$, $\bm{IC}(\overline{\mathcal{O}},\mathcal{L})$,
$\bm{IC}_{\mathcal{Q}}$: intersection cohomology complexes on orbit closures
$\overline{X}_{w},\overline{\mathcal{O}}$,
$\overline{\mathcal{Q}}=\mathcal{Q}$ (Equation 3.20, Equation 4.2, Equation
5.1)

$[\mathcal{A}]_{\oplus}$: split Grothendieck group of an additive category
$\mathcal{A}$

$\mathbb{H}_{H}^{\bullet}:D^{b}_{H}(X)\rightarrow H^{*}_{H}(X)\mathrm{-gmod}$:
equivariant hypercohomology functor

$\mathscr{D}=\\{(\mathcal{O},\mathcal{L})\mid\mathcal{O}\text{ is a $K$ orbit
on $X$},\mathcal{L}\in\operatorname{Loc}_{K}(\mathcal{O})\\}$: Lusztig–Vogan
parameters (3.1)

$M_{LV}$: the Lusztig–Vogan module of the Hecke algebra (Equation (3.2),
Definition 3.2.1, Lemma 3.2.3)

$M_{LV}^{0}$: trivial block of the Lusztig–Vogan module (Theorem 5.1.3)

$\mathcal{M}_{LV}^{ss}$: subcategory of semisimple complexes in $D^{b}_{K}(X)$
(Definition 4.1.1)

$\mathcal{M}_{LV}^{0}$: subcategory in $D^{b}_{K}(X)$ generated by trivial
local systems on closed orbits (Definition 5.2.1)

$\pi_{s}:X\rightarrow G/P_{s}$: projection to partial flag variety (3.4)

$\Theta_{s}=\pi_{s}^{*}\pi_{s*}[1]:D^{b}_{K}(X)\rightarrow D^{b}_{K}(X)$:
push-pull functor (4.3)

##  3\. Preliminaries

We begin by establishing our notation and conventions. We encourage readers
who are familiar with this material to skip to Section 4 and refer back to
this section only when needed.

###  3.1. The Hecke algebra

Let $(W,S)$ be the Coxeter system associated to the pair $(G,B)$. We denote by
$\ell:W\rightarrow\mathbb{N}$ the length function on $W$ and $\leq$ the Bruhat
order. A sequence $\underline{w}=(s_{1},s_{2},\ldots,s_{m})$ of simple
reflections $s_{i}\in S$ is called an expression for the group element $w$ if
$w=s_{1}s_{2}\cdots s_{m}\in W$. Notationally we distinguish between
expressions and group elements by underlining. We say an expression
$\underline{w}=(s_{1},s_{2},\ldots,s_{m})$ is reduced if $m=\ell(w)$.

The Hecke algebra $\bm{H}=\bm{H}(W,S)$ of the Coxeter system $(W,S)$ is the
unital associative $\mathbb{Z}[v^{\pm 1}]$-algebra generated by symbols
$\\{\delta_{s}\mid s\in S\\}$ subject to relations

  1. (i)

(quadratic) $\delta_{s}^{2}=1+(v^{-1}-v)\delta_{s}$ for all $s\in S$, and

  2. (ii)

(braid)
$\delta_{s}\delta_{t}\delta_{s}\ldots=\delta_{t}\delta_{s}\delta_{t}\ldots$
for all $s\neq t\in S$,

where the products in (ii) consist of $m_{st}$ elements on each side of the
equality.

######  Remark 3.1.1.

Alternatively, the Hecke algebra can be presented as the unital
$\mathbb{Z}[q^{\pm 1}]$-algebra generated by symbols $\\{T_{s}\mid s\in S\\}$
subject to relations

  1. (i)

$T_{s}^{2}=(q-1)T_{s}+q$ for all $s\in S$, and

  2. (ii)

$T_{s}T_{t}T_{s}\ldots=T_{t}T_{s}T_{t}\ldots$ for all $s\neq t\in S$,

where each product in (ii) consists of $m_{st}$ elements. Our initial
presentation can be obtained from this one by setting $v=q^{-1/2}$ and
$\delta_{s}=vT_{s}$.

The elements $\\{\delta_{s}\mid s\in S\\}$ are invertible, with inverses given
by $\delta_{s}^{-1}=\delta_{s}+(v-v^{-1})$. For $x\neq e\in W$, define an
element $\delta_{x}\in\bm{H}$ by choosing a reduced expression
$\underline{x}=(s_{1},\ldots,s_{m})$ of $x$ and setting
$\delta_{x}:=\delta_{s_{1}}\cdots\delta_{s_{m}}$. One can show that this
element is independent of choice of reduced expression. Define
$\delta_{e}:=1$. The set $\\{\delta_{x}\mid x\in W\\}$ is the standard basis
of $\bm{H}$.

The Hecke algebra comes equipped with a bar involution

$\bar{}:\bm{H}\rightarrow\bm{H},\hskip 5.69054pth\mapsto\bar{h}$

which is defined to be the unique ring homomorphism sending $v\mapsto v^{-1}$
and $\delta_{x}\mapsto\delta_{x^{-1}}$. We say an element $h\in\bm{H}$ is
self-dual if $\overline{h}=h$.

######  Theorem 3.1.2.

[KL79] [Soe97, Thm. 2.1] For each $x\in W$, there exists a unique self-dual
element $b_{x}\in\bm{H}$ such that
$b_{x}=\delta_{x}+\sum_{y<x}h_{y,x}\delta_{y}$ for $h_{y,x}\in
v\mathbb{Z}[v]$.

The set $\\{b_{x}\mid x\in W\\}$ is the Kazhdan–Lusztig basis of $\bm{H}$. The
polynomials $h_{y,x}\in v\mathbb{Z}[v]$ are Kazhdan–Lusztig polynomials. One
can check directly that the Kazhdan–Lusztig basis element corresponding to a
simple reflection $s\in S$ is $b_{s}=\delta_{s}+v$. The other Kazhdan–Lusztig
basis elements can be computed recursively using Theorem 3.1.2.

###  3.2. The Lusztig–Vogan module

The mathematical object at the heart of this project is a representation of
the Hecke algebra introduced in [LV83] and [Vog82], which we refer to as the
Lusztig–Vogan module. This representation is constructed using the geometry of
local systems on $K$-orbits in the flag variety of $G$. From this
representation one can extract character formulas for irreducible
representations of $G_{\mathbb{R}}$. In this section we sketch the
construction of Lusztig–Vogan. We refer the reader to [LV83] for full details
of the construction and [Vog82] for the relationship to representation theory
of $G_{\mathbb{R}}$.

Let $G$, $B$, $\theta$ and $K$ be as in Section 1.2. The first ingredient in
the construction of Lusztig–Vogan is a parameterizing set for a basis of their
$\bm{H}=\bm{H}(W,S)$-module. This set is defined geometrically, but it also
has significance in the representation theory of $G_{\mathbb{R}}$: it
parameterizes a certain set of equivalence classes of irreducible
representations of $G_{\mathbb{R}}$. See [Vog79a, Vog79b, Vog82, Vog83] for
the full story or [Mil93] for a geometric survey.

Let $X=G/B$ be the flag variety of $G$. The group $K$ acts on $X$ with
finitely many orbits [Mil93, Prop 5.1]. For a $K$-orbit $\mathcal{O}$, denote
by $\operatorname{Loc}_{K}(\mathcal{O})$ the set of irreducible
$K$-equivariant local systems on $\mathcal{O}$. Set

(3.1) $\mathscr{D}=\\{(\mathcal{O},\mathcal{L})\mid\mathcal{O}\text{ is a
$K$-orbit on $X$},\mathcal{L}\in\operatorname{Loc}_{K}(\mathcal{O})\\}.$

In a pair $(\mathcal{O},\mathcal{L})\in\mathscr{D}$, the local system
$\mathcal{L}$ determines the orbit $\mathcal{O}$, so we sometimes write
elements of $\mathscr{D}$ simply as $\mathcal{L}\in\mathscr{D}$.

Let $q$ be an indeterminate. Set

(3.2) $M_{LV}=\bigoplus_{\mathcal{L}\in\mathscr{D}}\mathbb{Z}[q^{\pm
1}]\mathcal{L}$

to be the free $\mathbb{Z}[q^{\pm 1}]$-module with basis $\mathscr{D}$.

The second ingredient in the construction is a set of endomorphisms
$\\{T_{s}\mid s\in S\\}$ of $M_{LV}$ which give it the structure of an
$\bm{H}$-module. These endomorphisms can be constructed in two ways: via push-
pull functors to partial flag varieties, or formulaically based on root type
of points in the orbit. Both constructions are equivalent. We will describe
the latter, and refer the reader to [LV83, §3] for the details on the former.

For a simple reflection $s\in S$, denote by

(3.3) $\displaystyle P_{s}=\text{ standard parabolic subgroup of $G$ of type
$s$},$ (3.4) $\displaystyle\pi_{s}:X\rightarrow G/P_{s}\text{ natural
projection map, and }$ (3.5) $\displaystyle
L_{x}^{s}=\pi_{s}^{-1}(\pi_{s}(x))\simeq\mathbb{P}^{1}\text{ the line of type
$s$ through $x\in X$}.$

Further, for a $K$-orbit $\mathcal{O}$, set

(3.6) $\widehat{\mathcal{O}}_{s}:=\bigcup_{x\in\mathcal{O}}L_{x}^{s}.$

######  Definition 3.2.1.

Let $(\mathcal{O},\mathcal{L})\in\mathscr{D}$, fix $x\in\mathcal{O}$, and use
notation (3.3) - (3.6). For $s\in S$, define $T_{s}:M_{LV}\rightarrow M_{LV}$
to be the endomorphism determined by the following formulas444We write the
endomorphism $T_{s}$ on the right to indicate that the corresponding action of
the Hecke algebra is a right action. (enumerated and labeled as to align with
[Vog82, Def. 6.4]).

  * (a)

(compact imaginary) If $L_{x}^{s}\subseteq\mathcal{O}$, then

$\mathcal{L}T_{s}=q\mathcal{L}.$

  * (b1)

(complex ascent) If $L_{x}^{s}\cap\mathcal{O}=\\{x\\}$ and
$\widehat{\mathcal{O}}\smallsetminus\mathcal{O}$ is a single $K$-orbit, then

$\mathcal{L}T_{s}=\widehat{\mathcal{L}}|_{\widehat{\mathcal{O}}\smallsetminus\mathcal{O}},$

where $\widehat{\mathcal{L}}$ is the unique extension (as a local system) of
$\mathcal{L}$ to $\widehat{\mathcal{O}}$.

  * (b2)

(complex descent) If
$L_{x}^{s}\cap\mathcal{O}=L_{x}^{s}\smallsetminus\\{\text{point}\\}$, then
$\widehat{\mathcal{O}}\smallsetminus\mathcal{O}$ is a single $K$-orbit and we
define

$\mathcal{L}T_{s}=(q-1)\mathcal{L}+q(\widehat{\mathcal{L}}|_{\widehat{\mathcal{O}}\smallsetminus\mathcal{O}}),$

where $\widehat{\mathcal{L}}$ is the unique extension of $\mathcal{L}$ to
$\widehat{\mathcal{O}}$.

  * (c1)

(type II noncompact imaginary) If
$L_{x}^{s}\cap\mathcal{O}=\\{x,x^{\prime}\\}$, then
$\widehat{\mathcal{O}}\smallsetminus\mathcal{O}$ is a single $K$-orbit and
$\mathcal{L}$ has two distinct extensions to $\widehat{\mathcal{O}}$, which we
denote $\widehat{\mathcal{L}}_{1}$ and $\widehat{\mathcal{L}}_{2}$. In this
case, we define

$\mathcal{L}T_{s}=\mathcal{L}+(\widehat{\mathcal{L}}_{1}+\widehat{\mathcal{L}}_{2})|_{\widehat{\mathcal{O}}\smallsetminus\mathcal{O}}.$

  * (c2)

(type II real) If
$L_{x}^{s}\cap\mathcal{O}=L_{x}^{s}\smallsetminus\\{\text{two points in one
$K$-orbit}\\}$ and $\mathcal{L}$ extends to $\widehat{\mathcal{O}}$, then

$\mathcal{L}T_{s}=(q-1)\mathcal{L}-\widehat{\mathcal{L}}_{2}|_{\mathcal{O}}+(q-1)\mathcal{L}^{\prime},$

where
$\mathcal{L}^{\prime}=\widehat{\mathcal{L}}|_{\widehat{\mathcal{O}}\smallsetminus\mathcal{O}}$
(here $\widehat{\mathcal{L}}$ is the extension of $\mathcal{L}$ to
$\widehat{\mathcal{O}}$) and $\widehat{\mathcal{L}}_{2}$ is the other
extension of $\mathcal{L}^{\prime}$ to $\widehat{\mathcal{O}}$.

  * (d1)

(type I noncompact imaginary) If $L_{x}^{s}\cap\mathcal{O}=\\{x\\}$ and
$\widehat{\mathcal{O}}\smallsetminus\mathcal{O}$ is the union of two
$K$-orbits $\mathcal{O}^{\prime}$ and $\mathcal{O}^{\prime\prime}$ (labeled so
that
$\dim\mathcal{O}=\dim\mathcal{O}^{\prime\prime}=\dim\mathcal{O}^{\prime}-1$),
then

$\mathcal{L}T_{s}=\widehat{\mathcal{L}}|_{\mathcal{O}^{\prime}}+\widehat{\mathcal{L}}|_{\mathcal{O}^{\prime\prime}},$

where $\widehat{\mathcal{L}}$ is the unique extension of $\mathcal{L}$ to
$\widehat{\mathcal{O}}$.

  * (d2)

(type I real) If $L_{x}^{s}\cap\mathcal{O}=L_{x}^{s}\smallsetminus\\{\text{two
points in two $K$-orbits $\mathcal{O}^{\prime}$ and
$\mathcal{O}^{\prime\prime}$}\\}$ and $\mathcal{L}$ extends to
$\widehat{\mathcal{O}}$, then

$\mathcal{L}T_{s}=(q-2)\mathcal{L}+(q-1)(\widehat{\mathcal{L}}|_{\mathcal{O}^{\prime}}+\widehat{\mathcal{L}}|_{\mathcal{O}^{\prime\prime}}),$

where $\widehat{\mathcal{L}}$ is the unique extension of $\mathcal{L}$ to
$\widehat{\mathcal{O}}$.

  * (e)

(real nonparity) If $L_{x}^{s}\cap\mathcal{O}=\\{\text{two points}\\}$ and
$\mathcal{L}$ does not extend to $\widehat{\mathcal{O}}$, then

$\mathcal{L}T_{s}=-\mathcal{L}.$

######  Remark 3.2.2.

A word on the labels in Definition 3.2.1: To an element
$(\mathcal{O},\mathcal{L})\in\mathscr{D}$, one can associate a $\theta$-stable
maximal torus $T$ of $G$ [Vog82, Cor. 2.2], which gives an action of $\theta$
on roots. The roots of $(G,B)$ can then be partitioned into “root types” based
on their behavior under this action [Vog82, Def. 2.5]. These root types are
the origin of the labels in Definition 3.2.1. These labels play no role in our
arguments, but we include them to orient any readers familiar with this
perspective.

######  Lemma 3.2.3.

[LV83, Cor. 3.5] For any $s\in S$, the endomorphism $T_{s}$ of Definition
3.2.1 satisfies

$T_{s}^{2}=(q-1)T_{s}+q.$

Hence the operators $\\{T_{s},s\in S\\}$ generate a right action of the Hecke
algebra $\bm{H}=\bm{H}(W,S)$ (in the formulation in Remark 3.1.1) on $M_{LV}$.

Definition 3.2.1 and Lemma 3.2.3 define the Lusztig–Vogan module $M_{LV}$. In
[LV83], several important properties of $M_{LV}$ are established. Most
notably, the existence and uniqueness of a self-dual (with respect to an
appropriately defined duality endomorphism on $M_{LV}$) basis for $M_{LV}$ is
proven. The change-of-basis matrix between this self-dual basis and the basis
$\mathscr{D}$ in (3.2) is given by a collection of polynomials which we will
refer to as Kazhdan–Lusztig–Vogan (KLV) polynomials. From these polynomials
one can extract character formulas for $G_{\mathbb{R}}$. We encourage the
uninitiated reader to explore [LV83] and [Vog82] in more detail to learn more
about this beautiful story.

######  Example 3.2.4.

$($The Lusztig–Vogan module of $SL_{2}(\mathbb{R}))$ Let
$G=SL_{2}(\mathbb{C})$ and $\theta:G\rightarrow G$ the involution given by

$\theta g=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}g\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$

Then $K=G^{\theta}$ is the torus of diagonal elements in $G$,

$K=\left\\{\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix}\mid
a\in\mathbb{C}^{\times}\right\\}\simeq\mathbb{C}^{\times}.$

The flag variety $X$ is isomorphic to $\mathbb{P}^{1}$, which we identify with
the set of lines through the origin in $\mathbb{C}^{2}$. Denote the line
through $(x,y)\in\mathbb{C}^{2}$ by $[x,y]$. The group $K$ acts on $X$ with
three orbits:

$\mathcal{Q}_{0}:=\\{[1,0]\\},\hskip
5.69054pt\mathcal{Q}_{\infty}:=\\{[0,1]\\},\text{ and }\hskip
2.84526pt\mathcal{O}:=\\{[x,y]\mid x\neq 0,y\neq
0\\}\simeq\mathbb{C}^{\times}.$

The stabilizers are $K$, $K$, and $\\{\pm I\\}$, respectively. As irreducible
$K$-equivariant local systems correspond to irreducible representations of the
component group of the stabilizer, we have four elements of $\mathscr{D}$:

$\mathscr{D}=\\{(\mathcal{Q}_{0},\mathbb{Q}_{\mathcal{Q}_{0}}),(\mathcal{Q}_{\infty},\mathbb{Q}_{\mathcal{Q}_{\infty}}),(\mathcal{O},\mathbb{Q}_{\mathcal{O}}),(\mathcal{O},\mathcal{L})\\},$

where $\mathcal{L}$ is the “Möbius band” local system corresponding to the
sign representation of $\\{\pm I\\}$.

Let $s\in S$ be the non-trivial simple reflection in $W=S_{2}$. We have
$\pi_{s}:X\rightarrow\mathrm{pt}$, and $L_{x}^{s}=X$ for any $x\in X$. An
examination of Definition 3.2.1 shows that the reflection $s$ is type I
noncompact imaginary (d1) with respect to the parameters
$(\mathcal{Q}_{0},\mathbb{Q}_{\mathcal{Q}_{0}})$ and
$(\mathcal{Q}_{\infty},\mathbb{Q}_{\mathcal{Q}_{\infty}})$, type I real (d2)
with respect to the parameter $(\mathcal{O},\mathbb{Q}_{\mathcal{O}})$, and
non-parity real (e) with respect to the parameter $(\mathcal{O},\mathcal{L})$.
Hence the action of the element $T_{s}\in\bm{H}$ on the basis of local systems
in $M_{LV}$ is given by the following formulas:

(3.7) $\displaystyle\mathbb{Q}_{\mathcal{Q}_{0}}\cdot
T_{s}=\mathbb{Q}_{\mathcal{Q}_{\infty}}+\mathbb{Q}_{\mathcal{O}},\hskip
19.91692pt\mathbb{Q}_{\mathcal{O}}\cdot
T_{s}=(q-2)\mathbb{Q}_{\mathcal{O}}+(q-1)(\mathbb{Q}_{\mathcal{Q}_{0}}+\mathbb{Q}_{\mathcal{Q}_{\infty}}),$
$\displaystyle\mathbb{Q}_{\mathcal{Q}_{\infty}}\cdot
T_{s}=\mathbb{Q}_{\mathcal{Q}_{0}}+\mathbb{Q}_{\mathcal{O}},\hskip
19.91692pt\mathcal{L}\cdot T_{s}=-\mathcal{L}.$

After extending scalars to $\mathbb{Z}[v^{\pm 1}]=\mathbb{Z}[q^{\pm 1/2}]$
(Remark 3.1.1), we can also describe the action of the Kazhdan–Lusztig basis
element $b_{s}=vT_{s}+v$ on the basis $\mathscr{D}$:

(3.8) $\displaystyle\mathbb{Q}_{\mathcal{Q}_{0}}\cdot
b_{s}=v(\mathbb{Q}_{\mathcal{Q}_{0}}+\mathbb{Q}_{\mathcal{Q}_{\infty}}+\mathbb{Q}_{\mathcal{O}}),\hskip
8.53581pt$ $\displaystyle\mathbb{Q}_{\mathcal{O}}\cdot
b_{s}=(v^{-1}-v)(\mathbb{Q}_{\mathcal{Q}_{0}}+\mathbb{Q}_{\mathcal{Q}_{\infty}}+\mathbb{Q}_{\mathcal{O}}),$
$\displaystyle\mathbb{Q}_{\mathcal{Q}_{\infty}}\cdot
b_{s}=v(\mathbb{Q}_{\mathcal{Q}_{0}}+\mathbb{Q}_{\mathcal{Q}_{\infty}}+\mathbb{Q}_{\mathcal{O}}),\hskip
5.69054pt$ $\displaystyle\mathcal{L}\cdot b_{s}=0.$

###  3.3. Soergel bimodules

Associated to the Coxeter system $(W,S)$ is a category of graded bimodules
called Soergel bimodules. The category of Soergel bimodules categorifies the
Hecke algebra of $(W,S)$. In this section, we describe the construction of
this category and establish our notation. For a detailed description of the
category of Soergel bimodules from both an algebraic and diagrammatic point of
view, see [EMTW20].

Fix a maximal torus $T$ of $G$ contained in $B$ and let

(3.9) $R:=S(X(T)_{\mathbb{Q}})$

be the symmetric algebra on
$X(T)_{\mathbb{Q}}:=X(T)\otimes_{\mathbb{Z}}\mathbb{Q}$, graded so that
$X(T)_{\mathbb{Q}}$ has degree $2$.

We will work in the category $R$-gbim of finitely generated (as both left and
right $R$-modules) graded $R$-bimodules, with morphisms given by graded
$R$-bimodule maps which are homogeneous of degree $0$. The category $R$-gbim
is the ambient home of Soergel bimodules. There is a natural shift functor
$(n):R\mathrm{-gbim}\rightarrow R\mathrm{-gbim}$ for each $n\in\mathbb{Z}$
sending $M\mapsto M(n)$, where $M(n)^{i}:=M^{n+i}$. Given an object $M$ in
$R$-gbim and a Laurent polynomial $p=\sum p_{i}v^{i}\in\mathbb{Z}_{\geq
0}[v^{\pm 1}]$, denote by

(3.10) $M^{\oplus p}:=\bigoplus M(i)^{\oplus p_{i}}.$

The tensor product $-\otimes_{R}-$ gives $R$-gbim the structure of a monoidal
category.

The natural action of $W$ on $X(T)_{\mathbb{Q}}$ extends to an action555See
Notation 6.2.2. of $W$ on $R$. For an element $s\in S$, denote by

$R^{s}=\\{r\in R\mid sr=r\\}.$

For $s\in S$, we define an object $B_{s}$ in $R$-gbim by

(3.11) $B_{s}:=R\otimes_{R^{s}}R(1).$

The modules $B_{s}$ for $s\in S$ are the building blocks of Bott–Samelson
bimodules.

######  Definition 3.3.1.

Let $\underline{w}=(s_{1},\ldots,s_{n})$ be an expression for $w\in W$. The
Bott–Samelson bimodule corresponding to $\underline{w}$ is the graded
$R$-bimodule

$BS(\underline{w}):=B_{s_{1}}\otimes_{R}B_{s_{2}}\otimes_{R}\cdots\otimes_{R}B_{s_{n}}.$

It is easy to see that there is a canonical isomorphism

$BS(\underline{w})=R\otimes_{R^{s_{1}}}R\otimes_{R^{s_{2}}}\cdots\otimes_{R^{s_{n}}}R(\ell(\underline{w})).$

With this we can define Soergel bimodules.

######  Definition 3.3.2.

Let $\mathbb{S}\text{Bim}$ be the additive Karoubian subcategory of $R$-gbim
generated by Bott-Samelson bimodules and their shifts. A Soergel bimodule is
an object in $\mathbb{S}\text{Bim}$. In other words, a Soergel bimodule is a
graded $R$-bimodule which is isomorphic to a direct summand of a finite direct
sum of grading shifts of Bott-Samelson bimodules.

The category $\mathbb{S}\text{Bim}$ of Soergel bimodules is closed under
tensor products, so it is a monoidal category. It is additive, but not
abelian.

######  Example 3.3.3.

$($Soergel bimodules of type $A_{1})$ Let $W=S_{2}$, generated by the simple
reflection $s$. The bimodules $R$ and $B_{s}$ are indecomposible Soergel
bimodules, and the square of $B_{s}$ is

(3.12) $\displaystyle B_{s}\otimes_{R}B_{s}$
$\displaystyle=R\otimes_{R^{s}}R\otimes_{R^{s}}R(2)$ $\displaystyle\simeq
R\otimes_{R^{s}}(R^{s}\oplus R^{s}(-2))\otimes_{R^{s}}R(2)$
$\displaystyle=B_{s}(1)\oplus B_{s}(-1).$

Hence up to shift and isomorphism, $R$ and $B_{s}$ are the only indecomposible
objects in $\mathbb{S}\text{Bim}$.

Denote by $[\mathbb{S}\text{Bim}]_{\oplus}$ the split Grothendieck group of
$\mathbb{S}\text{Bim}$. That is, $[\mathbb{S}\text{Bim}]_{\oplus}$ is the
abelian group generated by symbols $[M]$ for all objects
$M\in\mathbb{S}\text{Bim}$ subject to the relations
$[M]=[M]^{\prime}+[M^{\prime\prime}]$ whenever $M\simeq M^{\prime}\oplus
M^{\prime\prime}$ in $\mathbb{S}\text{Bim}$. We can give
$[\mathbb{S}\text{Bim}]_{\oplus}$ the structure of a $\mathbb{Z}[v^{\pm
1}]$-module by

$p[M]:=[M^{\oplus p}]$

for $M\in\mathbb{S}\text{Bim}$ and $p\in\mathbb{Z}_{\geq 0}[v^{\pm 1}]$, with
notation as in (3.10). Because $\mathbb{S}\text{Bim}$ is monoidal,
$[\mathbb{S}\text{Bim}]_{\oplus}$ has the structure of a $\mathbb{Z}[v^{\pm
1}]$-algebra.

Soergel showed that $\mathbb{S}\text{Bim}$ categorifies the Hecke algebra.

######  Theorem 3.3.4.

(Soergel’s categorification theorem) The map $b_{s}\mapsto[B_{s}]$ defines an
isomorphism of $\mathbb{Z}[v^{\pm 1}]$-algebras

$\bm{H}\xrightarrow{\sim}[\mathbb{S}\text{Bim}]_{\oplus}.$

###  3.4. The equivariant derived category

The geometric setting of this paper is the equivariant derived category, as
developed in [BL94]. Here we briefly recall some of its basic structure.

Let $H$ be a complex algebraic group and $X$ a $H$-variety (over $\mathbb{C}$,
equipped with the classical topology). Denote by
$D_{c}^{b}(X):=D^{b}_{c}(X;\mathbb{Q})$ the bounded derived category of
constructible (with respect to the $H$-orbit stratification) sheaves of
$\mathbb{Q}$-vector spaces on $X$. Fix a smooth model $EH\rightarrow BH$ for
the classifying space of $H$, as in [BL94, §12.4.1]. Let

$X\xleftarrow{p}X\times EH\xrightarrow{q}X\times_{H}EH$

be the natural projections. We denote elements in quotient spaces using
brackets; e.g., we write $[x,e]\in X\times_{H}EH$ for $x\in X$ and $e\in EH$.
Denote by $D^{b}(X\times EH)$ and $D^{b}(X\times_{H}EH)$ the bounded derived
categories of $\mathbb{Q}$-vector spaces on $X\times EH$ and $X\times_{H}EH$,
respectively. The (bounded, constructible) equivariant derived category
$D^{b}_{H}(X):=D^{b}_{H}(X;\mathbb{Q})$ is defined as follows.

######  Definition 3.4.1.

An object $\mathcal{G}\in D^{b}_{H}(X)$ is a triple
$\mathcal{G}=(\mathcal{G}_{X},\overline{\mathcal{G}},\beta_{\mathcal{G}})$
with $\mathcal{G}_{X}\in D^{b}_{c}(X)$, $\overline{\mathcal{G}}\in
D^{b}(X\times_{H}EH)$, and
$\beta_{\mathcal{G}}:p^{*}\mathcal{G}_{X}\xrightarrow{\sim}q^{*}\overline{\mathcal{G}}$
an isomorphism in $D^{b}(X\times_{H}EH)$. A morphism
$\varphi:\mathcal{G}\rightarrow\mathcal{F}$ in $D_{H}^{b}(X)$ is a pair
$\varphi=(\varphi_{X},\overline{\varphi})$ with
$\varphi_{X}:\mathcal{G}_{X}\rightarrow\mathcal{F}_{X}$ a morphism in
$D^{b}_{c}(X)$ and
$\overline{\varphi}:\overline{\mathcal{G}}\rightarrow\overline{\mathcal{F}}$ a
morphism in $D^{b}(X\times_{H}EH)$ such that the diagram

${p^{*}\mathcal{G}_{X}}$${p^{*}\mathcal{F}_{X}}$${q^{*}\overline{\mathcal{G}}}$${q^{*}\overline{\mathcal{F}}}$$\scriptstyle{p^{*}\varphi_{X}}$$\scriptstyle{\beta_{\mathcal{G}}}$$\scriptstyle{\beta_{\mathcal{F}}}$$\scriptstyle{q^{*}\overline{\varphi}}$

commutes.

There is a forgetful functor

(3.13) $\mathrm{For}:D^{b}_{H}(X)\rightarrow D^{b}(X)$

sending
$\mathcal{G}=(\mathcal{G}_{X},\overline{\mathcal{G}},\beta_{\mathcal{G}})$ to
$\operatorname{For}\mathcal{G}:=\mathcal{G}_{X}$. Let $\mathbb{Q}_{X}\in
D^{b}_{c}(X)$ be the constant sheaf, viewed as a complex concentrated in
degree zero. There is a canonical equivariant lift
$(\mathbb{Q}_{X},\mathbb{Q}_{X\times_{H}EH},id)\in D^{b}_{H}(X)$ of
$\mathbb{Q}_{X}$ which we will also denote by $\mathbb{Q}_{X}$. Whether we are
referring to the object of $D^{b}_{H}(X)$ or $D^{b}_{c}(X)$ should always be
clear from context.

The equivariant derived category $D^{b}_{H}(X)$ is triangulated and supports
the usual six functor formalism in a way that is compatible with the forgetful
functor. It also supports a natural shift functor

$[n]:\mathcal{G}\mapsto\mathcal{G}[n]:=(\mathcal{G}_{X}[n],\overline{\mathcal{G}}[n],\beta_{\mathcal{G}}[n])$

for each $n\in\mathbb{Z}$. Given $\mathcal{G},\mathcal{F}\in D^{b}_{H}(X)$, we
denote by

(3.14)
$\operatorname{Hom}_{H}(\mathcal{G},\mathcal{F}):=\operatorname{Hom}_{D^{b}_{H}(X)}(\mathcal{G},\mathcal{F}),$

and

(3.15)
$\operatorname{Hom}^{\bullet}_{H}(\mathcal{G},\mathcal{F}):=\bigoplus_{n\in\mathbb{Z}}\operatorname{Hom}_{D^{b}_{H}(X)}(\mathcal{G},\mathcal{F}[n]).$

There are two cohomology functors on $D^{b}_{H}(X)$: the equivariant
hypercohomology functor

(3.16)
$\mathbb{H}^{\bullet}_{H}:=\operatorname{Hom}^{\bullet}_{H}(\mathbb{Q}_{X},-):D^{b}_{H}(X)\rightarrow
H^{*}_{H}(X)\mathrm{-gmod}$

and the ordinary hypercohomology functor

(3.17)
$\mathbb{H}^{\bullet}:=\operatorname{Hom}^{\bullet}_{D^{b}_{c}(X)}(\mathbb{Q}_{X},\operatorname{For}(-)):D^{b}_{H}(X)\rightarrow
H^{*}(X)\mathrm{-gmod}.$

Here $H^{*}_{H}(X)-$gmod (resp. $H^{*}(X)-$gmod) is the category of graded
modules over the graded ring $H^{*}_{H}(X):=H^{*}(X\times_{H}EH)$ (resp.
$H^{*}(X)$). The graded vector space
$\operatorname{Hom}_{H}^{\bullet}(\mathbb{Q}_{X},\mathcal{G})$ (resp.
$\operatorname{Hom}^{\bullet}_{D^{b}_{c}(X)}(\mathbb{Q}_{X},\mathcal{G})$)
gains a
$H^{*}_{H}=\operatorname{Hom}^{\bullet}_{H}(\mathbb{Q}_{X},\mathbb{Q}_{X})$-action
(resp.
$H^{*}(X)=\operatorname{Hom}^{\bullet}_{D^{b}_{c}(X)}(\mathbb{Q}_{X},\mathbb{Q}_{X})$-action)
via precomposition.

The hypercohomology functors agree with the usual sheaf cohomology functors:
for every $\mathcal{G}\in D^{b}_{H}(X)$,

(3.18)
$\mathbb{H}_{H}^{\bullet}(\mathcal{G})=H^{*}(X\times_{H}EH;\overline{\mathcal{G}}),\text{
and }\mathbb{H}^{\bullet}(\mathcal{G})=H^{*}(X;\mathcal{G}_{X}).$

Equivariant hypercohomology can be computed using the Leray spectral sequence
associated to the fibration $X\times_{H}EH\rightarrow BH$, which simplifies to

(3.19)
$E^{pq}_{2}=H^{p}(BH)\otimes\mathbb{H}^{q}(\mathcal{G})\implies\mathbb{H}^{p+q}_{H}(\mathcal{G}).$

###  3.5. The geometric Hecke category

The definitions of Section 3.3 have their roots in geometry. There is another
categorification of the Hecke algebra which can be found within the
equivariant derived category $D^{b}_{B}(G/B)$, coming from Grothendieck’s
function-sheaf dictionary. This geometric incarnation of the Hecke category
will play a key role in our arguments that follow, so we recall its
construction. For more details (including a more general framework which makes
the function-sheaf perspective explicit), see [Wil18, §2].

Return to the setup of Section 1, and let $X:=G/B$ be the flag variety of $G$.
The action of $B$ on $X$ stratifies $X$ into Schubert cells:

$X=\bigsqcup_{w\in W}X_{w}\text{ where
}X_{w}:=BwB/B\simeq\mathbb{A}^{\ell(w)}.$

Let $D^{b}_{B}(X):=D^{b}_{B}(X;\mathbb{Q})$ be the $B$-equivariant derived
category, as in Section 3.4. The category $D^{b}_{B}(X)$ has a monoidal
structure given by convolution, which is constructed as follows. Denote by
$\mathrm{mult}:G\times_{B}X\rightarrow X$ the map induced by multiplication in
$G$. The convolution of $\mathcal{F},\mathcal{G}\in D^{b}_{B}(X)$ is defined
to be

$\mathcal{F}*\mathcal{G}:=\mathrm{mult}_{*}(\mathcal{F}\boxtimes_{B}\mathcal{G}),$

where $\mathcal{F}\boxtimes_{B}\mathcal{G}$ is obtained by descent from
$\mathcal{F}\boxtimes\mathcal{G}\in D^{b}_{B^{3}}(G\times X)$.

For $s\in S$, let

$P_{s}:=\overline{BsB}=BsB\sqcup B$

be the corresponding parabolic subgroup of $G$. Denote by
$\mathbb{Q}_{P_{s}/B}$ the constant sheaf on $P_{s}/B$, considered as a sheaf
on $X$ by extending by zero.

######  Definition 3.5.1.

The geometric Hecke category $\mathcal{H}$ is the full subcategory of
$D^{b}_{B}(X)$ generated by $\\{\mathbb{Q}_{P_{s}/B}\\}_{s\in S}$ under
convolution ($*$), direct sums ($\oplus$), homological shifts ($[1]$), and
direct summands ($\ominus$):

$\mathcal{H}=\langle\mathbb{Q}_{P_{s}/B}\mid s\in
S\rangle_{*,\oplus,[1],\ominus}.$

By construction, $\mathcal{H}$ is closed under convolution, so it is a
monoidal category. It is additive, but not abelian.

Let $[\mathcal{H}]_{\oplus}$ be the split Grothendieck group666See end of
Section 3.3 for a definition of split Grothendieck group. of $\mathcal{H}$. We
can give $[\mathcal{H}]_{\oplus}$ the structure of a $\mathbb{Z}[v^{\pm
1}]$-module by setting

$v\cdot[\mathcal{F}]:=[\mathcal{F}[1]].$

The monoidal structure on $\mathcal{H}$ gives $[\mathcal{H}]_{\oplus}$ the
structure of a $\mathbb{Z}[v^{\pm 1}]$-algebra. As one might hope from its
name, the geometric Hecke category categorifies the Hecke algebra.

######  Theorem 3.5.2.

The map $b_{s}\mapsto[\mathbb{Q}_{P_{s}/B}[1]]$ defines an isomorphism of
$\mathbb{Z}[v^{\pm 1}]$-algebras

$\bm{H}\xrightarrow{\sim}[\mathcal{H}]_{\oplus}.$

For a proof, see, for example [Wil18].

###  3.6. From the geometric Hecke category to Soergel bimodules

By Theorem 3.3.4 and Theorem 3.5.2, both $\mathbb{S}\text{Bim}$ and
$\mathcal{H}$ categorify $\bm{H}$. In fact, their relationship can be made
more explicit using an alternate characterization of the geometric Hecke
category.

The category $D^{b}_{B}(X)$ is Krull-Schmidt, meaning that every object
decomposes into a finite direct sum of indecomposable objects, which are
characterized by having local endomorphism rings. Hence, to understand the
category $\mathcal{H}$, we should start by understanding the indecomposible
objects. By definition, objects in $\mathcal{H}$ are direct summands of finite
direct sums of shifts of objects of the form

$\mathcal{E}_{\underline{w}}:=\mathbb{Q}_{P_{s_{1}}/B}*\mathbb{Q}_{P_{s_{2}}/B}*\cdots*\mathbb{Q}_{P_{s_{n}}/B}$

for some expression $\underline{w}=(s_{1},s_{2},\ldots,s_{n})$ of an element
$w\in W$. The objects $\mathcal{E}_{\underline{w}}$ can be described in terms
of Bott-Samelson resolutions of the Schubert varieties $\overline{X}_{w}$. If
$\underline{w}=(s_{1},s_{2},\ldots,s_{n})$ is an expression for $w\in W$, then
the Bott-Samelson space

$BS_{\underline{w}}:=P_{s_{1}}\times_{B}P_{s_{2}}\times_{B}\cdots\times_{B}P_{s_{n}}/B$

is a smooth algebraic variety. There is a natural projective morphism

$\xi:BS_{\underline{w}}\rightarrow\overline{X}_{w}$

induced by multiplication in $G$. For any expression $\underline{w}$, there is
a canonical isomorphism [Soe00, Lem. 3.2.1]

$\xi_{*}\mathbb{Q}_{BS_{\underline{w}}}=\mathcal{E}_{\underline{w}}.$

######  Remark 3.6.1.

If $\underline{w}$ is a reduced expression for $w\in W$, then $\xi$ is a
resolution of singularities of $\overline{X}_{w}$. Such resolutions are
sometimes called Bott-Samelson resolutions.

Let

(3.20) $\bm{IC}_{w}:=\bm{IC}(\overline{X}_{w})\in D^{b}_{B}(G/B)$

denote the $B$-equivariant intersection cohomology complex on
$\overline{X}_{w}$, shifted by $\ell(w)$, extended by zero, so it is perverse
on $X$. The morphism $\xi$ is proper, so by the decomposition theorem,
$\mathcal{E}_{\underline{w}}$ decomposes into a finite direct sum of shifts of
$\bm{IC}_{w}$ for $w\in W$. This gives us a second characterization of
$\mathcal{H}$:

(3.21) $\mathcal{H}=\langle\bm{IC}_{w}\mid w\in W\rangle_{\oplus,[1]}.$

Using the characterization (3.21), the relationship between $\mathcal{H}$ and
$\mathbb{S}\text{Bim}$ becomes clear by applying hypercohomology. Let

(3.22) $\mathbb{H}_{B}^{\bullet}:D^{b}_{B}(X)\rightarrow
H^{*}_{B}(X)-\mathrm{gmod}$

be the equivariant hypercohomology functor, as in (3.16). There is a canonical
isomorphism of graded algebras

$H^{*}_{B}(X)=H^{*}_{T}(X)=R\otimes_{R^{W}}R,$

where $R=S(\mathfrak{t}^{*})$ as in (3.9) (cf. [Bri98, Prop. 1(iii)]). The
quotient map $R\otimes R\rightarrow R\otimes_{R^{W}}R$ induces a fully
faithful functor

(3.23) $R\otimes_{R^{W}}R\mathrm{-gmod}\rightarrow R\mathrm{-gbim}.$

By identifying $H_{B}^{*}(X)$ with its image under (3.23), we can rewrite
(3.22) as

$\mathbb{H}^{\bullet}_{B}:D^{b}_{B}(X)\rightarrow R\mathrm{-gbim}.$

Soergel observed that when restricted to the geometric Hecke category, this
functor is fully faithful, and its essential image is the category of Soergel
bimodules. Hence $B$-equivariant hypercohomology provides an equivalence

(3.24)
$\mathbb{H}^{\bullet}_{B}:\mathcal{H}\xrightarrow{\sim}\mathbb{S}\text{Bim}.$

Under the equivalence (3.24), we can see the players of Section 3.3 in their
geometric guise:

(3.25) $\displaystyle\mathbb{H}^{\bullet}_{B}(\bm{IC}_{s})$
$\displaystyle=B_{s}\text{ for all }s\in S,\text{ and }$ (3.26)
$\displaystyle\mathbb{H}^{\bullet}_{B}(\mathcal{E}_{\underline{w}})$
$\displaystyle=BS(\underline{w})\text{ for all expressions
}\underline{w}\text{ of all elements }w\in W.$

######  Remark 3.6.2.

Equation (3.26) together with Remark 3.6.1 explains the terminology “Bott-
Samelson bimodule” in Definition 3.3.1.

##  4\. A geometric categorification

In this section, we use the equivariant derived category $D_{K}^{b}(X)$ to
give a geometric categorification of the Lusztig–Vogan module (q.v. Theorem
4.1.3), we study convolution of complexes of sheaves via character maps to the
Lusztig–Vogan module, and we compute cohomology of all fibers for any
resolution constructed in [Lar20] (q.v. Theorem 4.4.1). On one hand, our
geometric categorification can be viewed as studying equivariant proper maps
via linearization, and on the other hand, it is a stepping stone to our
algebraic categorification in Section 6.

Recall that $K$-equivariant local systems on $\mathcal{O}$ are in bijection
with representations of the component group $K_{x}/K_{x}^{0}$ of the
stabilizer $K_{x}$ of $x\in\mathcal{O}$ in $K$. Irreducible representations
correspond to irreducible local systems. Given an irreducible $K$-equivariant
local system $\mathcal{L}\in\operatorname{Loc}_{K}(\mathcal{O})$, denote by

(4.1) $\widetilde{\mathcal{L}}\in D_{K}^{b}(X)$

the intermediate extension of $\mathcal{L}$ to $\overline{\mathcal{O}}$
extended by zero to $X$. Set

(4.2)
$\bm{IC}(\overline{\mathcal{O}},\mathcal{L})=\widetilde{\mathcal{L}}[\dim(\mathcal{O})]$

to be the corresponding self-dual object.

###  4.1. Semisimple complexes

We study a subcategory of the $K$-equivariant derived category. The category
consists of all semisimple complexes in $D_{K}^{b}(X)$, which is the
$K$-equivariant analogue of the geometric Hecke category (q.v. (3.21)).

######  Definition 4.1.1.

Let

$\mathcal{M}_{LV}^{ss}:=\langle\bm{IC}(\overline{\mathcal{O}},\mathcal{L})\mid(\mathcal{O},\mathcal{L})\in\mathscr{D}\rangle_{\oplus,[1]}$

be the subcategory of $D^{b}_{K}(X)$ generated by $\bm{IC}$ sheaves under the
operations of taking direct sums and grading shifts.

By construction, the category $\mathcal{M}_{LV}^{ss}$ is an additive category.
It also admits a natural categorical action (in the sense of [EGNO16, Ch. 7])
of the monoidal category $\mathcal{H}$ via convolution. The convolution
formalism described in Section 3.5 gives a right action of $D^{b}_{B}(X)$ on
$D^{b}_{K}(X)$: for $\mathcal{F}\in D^{b}_{K}(X)$ and $\mathcal{G}\in
D^{b}_{B}(X)$, define

$\mathcal{F}*\mathcal{G}=\mathrm{mult}_{*}(\mathcal{F}\boxtimes_{B}\mathcal{G}),$

where $\mathrm{mult}:G\times_{B}X\rightarrow X$ is as in Section 3.5 and
$\mathcal{F}\boxtimes_{B}\mathcal{G}$ is obtained via descent from
$\mathcal{F}\boxtimes\mathcal{G}\in D^{b}_{K\times B\times B}(G\times X)$. By
restricting this action to the geometric Hecke category $\mathcal{H}$, we
obtain an action of $\mathcal{H}$ on $D^{b}_{K}(X)$.

The convolution action of $\mathcal{H}$ on $D^{b}_{K}(X)$ can also be realized
in terms of push-pull functors. For $s\in S$, define a functor777We write
$\mathcal{F}\Theta_{s}$ for the image of $\mathcal{F}$ under the functor
$\Theta_{s}$ to remind ourselves that we should think of the functors
$\Theta_{s}$, $s\in S$ as providing a right action of $\mathcal{H}$ on
$D^{b}_{K}(X)$. $\Theta_{s}:D^{b}_{K}(X)\rightarrow D^{b}_{K}(X)$ by

(4.3) $\mathcal{F}\Theta_{s}:=\pi_{s}^{*}\pi_{s*}\mathcal{F}[1]$

for $\mathcal{F}\in D^{b}_{K}(X)$. As in (3.20), denote by
$\bm{IC}_{s}:=\bm{IC}(P_{s}/B)\in\mathcal{H}$.

######  Lemma 4.1.2.

For any object $\mathcal{F}\in D_{K}^{b}(X)$ and $s\in S$,

$\mathcal{F}\Theta_{s}\simeq\mathcal{F}*\bm{IC}_{s}.$

###### Proof.

See [Soe90, LV83]. ∎

Because the map $\pi_{s}$ is proper for any $s\in S$, the decomposition
theorem stipulates that for $(\mathcal{O},\mathcal{L})\in\mathscr{D}$, the
complex $\bm{IC}(\overline{\mathcal{O}},\mathcal{L})\Theta_{s}$ decomposes
into a direct sum of shifts of $\bm{IC}$ sheaves. In particular,

$\bm{IC}(\overline{\mathcal{O}},\mathcal{L})\Theta_{s}\in\mathcal{M}_{LV}^{ss}\text{
for any }(\mathcal{O},\mathcal{L})\in\mathscr{D}\text{ and }s\in S.$

This shows that the category $\mathcal{M}_{LV}^{ss}\subseteq D^{b}_{K}(X)$ is
stable under the action of $\mathcal{H}$.

For technical reasons we extend scalars of $M_{LV}$ to $\mathbb{Z}[q^{\pm
1/2}]$. This will enable odd shifts of $\widetilde{\mathcal{L}}$ in our
category; we do this to include all semisimple complexes, and in particular,
self-dual complexes. Given a $K$-orbit $\mathcal{O}$, let
$k_{\mathcal{O}}\,\colon\mathcal{O}\to X$ be the locally closed inclusion. By
construction, the cohomology sheaves
$\mathcal{H}^{n}(k_{\mathcal{O}}^{\ast}\mathcal{F})$ are objects in the
abelian category of $K$-equivariant constructible sheaves on $\mathcal{O}$.
For an irreducible $K$-equivariant local system $\mathcal{L}$ on
$\mathcal{O}$, denote by

(4.4) $[\mathcal{L}\colon\mathcal{H}^{n}(k_{\mathcal{O}}^{\ast}\mathcal{F})]$

the multiplicity of $\mathcal{L}$ in
$\mathcal{H}^{n}(k_{\mathcal{O}}^{\ast}\mathcal{F})$. Define
$\operatorname{ch}\,\colon D_{K}^{b}(X)\to M_{LV}$ by

(4.5)
$\mathcal{F}\mapsto\sum_{n,(\mathcal{O},\mathcal{L})}[\mathcal{L}:\mathcal{H}^{n}(k_{\mathcal{O}}^{\ast}\mathcal{F})]\,q^{n/2}\,\mathcal{L}.$

The character map $\operatorname{ch}$ does not intertwine Hecke actions on
local systems, but we show in Section 4.3 that $\operatorname{ch}$ does
intertwine the Hecke actions when restricted to our semisimple category.

######  Theorem 4.1.3.

Let $\mathcal{M}_{LV}^{ss}$ be as in Definition 4.1.1 and $M_{LV}$ the
Lusztig–Vogan module (Equation (3.2), Definition 3.2.1, Lemma 3.2.3). As right
$\bm{H}$-modules,

$[\mathcal{M}_{LV}^{ss}]_{\oplus}\simeq M_{LV}.$

We prove Theorem 4.1.3 in Section 4.3 by relating $\operatorname{ch}$ to an
intermediary character map and reducing computations to root types. The
formula for each root type will be seen to agree with the formulas in
Definition 3.2.1. We begin by analyzing local systems.

###  4.2. Local systems

We start by introducing an auxiliary character map. Define $m\,\colon
D_{K}^{b}(X)\to M_{LV}$ by

(4.6)
$\mathcal{F}\mapsto\sum_{n,(\mathcal{O},\mathcal{L})}(-1)^{n}[\mathcal{L}:\mathcal{H}^{n}(k^{\ast}_{\mathcal{O}}\mathcal{F})]\,q^{\lfloor
n/2\rfloor}\,\mathcal{L},$

where $\lfloor n\rfloor$ is the floor function. We include some easy facts
about (4.6).

  * •

For every $\mathcal{F}\in D_{K}^{b}(G/B)^{\ast-even}$ (q.v. Definition 7.2.8)
we have

(4.7) $m(\mathcal{F})=\operatorname{ch}(\mathcal{F}).$

  * •

For every $\mathcal{F}\in D_{K}^{b}(X)$ we have

(4.8) $m(\mathcal{F}[-2])=qm(\mathcal{F}).$

  * •

If $\mathcal{H}^{\ast}(\mathcal{F})\in D_{K}^{b}(X)$ denotes the direct sum of
the cohomology sheaves of $\mathcal{F}$, then we have

(4.9) $m(\mathcal{F})=m(\mathcal{H}^{\ast}(\mathcal{F})).$

  * •

If $\mathcal{E}\to\mathcal{F}\to\mathcal{G}\overset{+1}{\longrightarrow}$ is a
distinguished triangle in $D_{K}^{b}(X)$ such that $\mathcal{G}\in
D_{K}^{b}(X)^{\ast-even}$, then

(4.10) $m(\mathcal{F})=m(\mathcal{E})+m(\mathcal{G}).$

######  Lemma 4.2.1.

For every $(\mathcal{O},\mathcal{L})\in\mathscr{D}$ and $s\in S$, we have

(4.11)
$m(\pi_{s}^{\ast}\pi_{s\ast}k_{\mathcal{O}!}\mathcal{L})=\mathcal{L}(T_{s}+1).$

The proof is a case-by-case analysis of root types provided below. We require
lemmata before proceeding.

For every $s\in S$ and $x\in X$ define

(4.12)
${KxB\times_{B}BsB/B}$${KxBsB/B}$${KxB/B}$$\scriptstyle{\mu}$$\scriptstyle{\pi}$

to be the natural projection and multiplication maps. For every $y\in KxBsB/B$
we have

(4.13) $\mu^{-1}(y)=KxB/B\cap yBsB/B.$

######  Lemma 4.2.2.

For every $(\mathcal{O},\mathcal{L})\in\mathscr{D}$ and $s\in S$, the relation
(4.11) holds true if and only if

(4.14) $m(k_{!}\mu_{!}\pi^{\ast}\mathcal{L})=\mathcal{L}T_{s},$

where $k\,\colon KxBsB/B\to X$ is the locally closed inclusion, and $\mu$ and
$\pi$ are as in (4.12).

###### Proof.

The diagram

(4.15)
${KxB\times_{B}P_{s}/B}$${KxP_{s}/B}$${G/B}$${KxB/B}$${KxP_{s}/P_{s}}$${G/P_{s}}$$\scriptstyle{\bar{\mu}}$$\scriptstyle{\bar{\pi}}$$\scriptstyle{\pi_{s}^{\prime}}$$\scriptstyle{k^{\prime\prime}}$$\scriptstyle{\pi_{s}}$$\scriptstyle{\widetilde{\pi}_{s}}$$\scriptstyle{\ell}$

is cartesian so by proper base change

(4.16)
$\begin{split}\pi_{s}^{\ast}\pi_{s\ast}k_{\mathcal{O}!}&=\pi_{s}^{\ast}(\pi_{s}k_{\mathcal{O}})_{!}\\\
&=\pi_{s}^{\ast}(\ell\widetilde{\pi}_{s})_{!}\\\
&=(k^{\prime\prime}\bar{\mu})_{!}\bar{\pi}^{\ast}\end{split}$

holds true. Let

(4.17) $\begin{split}i\,\colon KxB/B&\to KxB\times_{B}P_{s}/B\\\ j\,\colon
KxB\times_{B}BsB/B&\to KxB\times_{B}P_{s}/B\end{split}$

be complementary closed and open inclusions. The distinguished triangle

(4.18)
$j_{!}j^{\ast}\bar{\pi}^{\ast}\mathcal{L}\to\bar{\pi}^{\ast}\mathcal{L}\to
i_{\ast}i^{\ast}\bar{\pi}^{\ast}\mathcal{L}\overset{+1}{\longrightarrow}$

gives the distinguished triangle

(4.19)
$(k\mu)_{!}\pi^{\ast}\mathcal{L}\to\pi_{s}^{\ast}(\pi_{s}k_{\mathcal{O}})_{!}\mathcal{L}\to
k_{\mathcal{O}!}\mathcal{L}\overset{+1}{\longrightarrow}$

under the triangulated functor $(k^{\prime\prime}\bar{\mu})_{!}$. The
character map $m$ (4.6) is additive on (4.19) by (4.10). Because
$m(k_{\mathcal{O}!}\mathcal{L})=\mathcal{L}$, we have

(4.20)
$m(\pi_{s}^{\ast}(\pi_{s}k_{\mathcal{O}})_{!}\mathcal{L})=m((k\mu)_{!}\pi^{\ast}\mathcal{L})+\mathcal{L}.$

The result follows. ∎

######  Lemma 4.2.3.

Let $\zeta\,\colon Z\to Y$ be a locally trivial fiber bundle with fiber
$F=F_{y}=\zeta^{-1}(y)$. Assume that for every $n\in\mathbb{Z}$ there are
finitely many elements $f_{n\ell}\in H^{n}(Z)$ such that for every $y\in Y$ we
obtain a $\mathbb{Q}$-basis for $H^{n}(F_{y})$ by restriction. Let
$\zeta^{\prime}\,\colon Z^{\prime}\to Y^{\prime}$ be the pullback of $\zeta$
along a map $Y^{\prime}\to Y$. Then for every $p\in\mathbb{Z}$, every
$z^{\prime}\in H^{p}(Z^{\prime})$ can be written uniquely as

(4.21) $z^{\prime}=\sum_{n\ell}y_{n\ell}^{\prime}f_{n\ell},$

for some $y_{n\ell}^{\prime}\in H^{p-n}(Y^{\prime})$.

###### Proof.

See, e.g., [AF21, App. A §4] for the case of identity map $Y^{\prime}\to Y$
(and $Z^{\prime}=Z$). Let $y^{\prime}\mapsto y$ so we have the canonical
homeomorphism $F_{y^{\prime}}\to F_{y}$ as usual. The commutative diagram

(4.22)
${F_{y^{\prime}}}$${F_{y}}$${Z^{\prime}}$${Z}$$\scriptstyle{\operatorname{\cong}}$

shows that pulling back $f_{ij}$ to $f_{ij}^{\prime}\in H^{n}(Z^{\prime})$
restricts to a basis in $H^{n}(F_{y^{\prime}})$. Applying the case of
$Z^{\prime}=Z$ to $Z^{\prime}$ proves the lemma. ∎

######  Lemma 4.2.4.

Let $\zeta\,\colon Z\to Y$ be a smooth map with fiber $F=F_{y}=\zeta^{-1}(y)$.
Assume that for every $n\in\mathbb{Z}$ there are finitely many elements
$f_{n\ell}\in H^{n}(Z)$ such that for every $y\in Y$ we obtain a
$\mathbb{Q}$-basis for $H^{n}(F_{y})$ by restriction. For every
$\mathcal{F}\in D^{b}(Y)$ we have

(4.23) $\zeta_{!}\zeta^{\ast}\mathcal{F}\cong\mathcal{F}\boxtimes
H_{c}^{\ast}(F).$

###### Proof.

We have

(4.24)
$\zeta_{!}\zeta^{\ast}\mathcal{F}\operatorname{\cong}\mathcal{F}\underset{\mathbb{Q}_{Y}}{\otimes}\zeta_{!}\mathbb{Q}_{Z}$

by [KS94, Prop. 2.6.6], so it is enough to prove (4.23) for the constant
sheaf. Identify $\zeta=\zeta\times\operatorname{id}_{\operatorname{pt}}$. Then

(4.25)
$\mathcal{H}om(\zeta_{!}\zeta^{\ast}\mathbb{Q}_{Y},\mathbb{Q}_{Y}\boxtimes
H_{c}^{\ast}(F))\operatorname{\cong}\zeta_{\ast}\mathcal{H}om(\zeta^{\ast}\mathbb{Q}_{Y},\zeta^{!}\mathbb{Q}_{Y}\boxtimes
H_{c}^{\ast}(F))$

by Verdier duality. For smooth morphisms the relation

(4.26) $\zeta^{!}\mathbb{Q}_{Y}\operatorname{\cong}\mathbb{Q}_{Z}[2d]$

is a relative Poincaré duality. It follows that

(4.27)
$\mathcal{H}om(\zeta_{!}\zeta^{\ast}\mathbb{Q}_{Y},\mathbb{Q}_{Y}\boxtimes
H_{c}^{\ast}(F))\operatorname{\cong}\zeta_{\ast}\mathcal{H}om(\mathbb{Q}_{Z},\mathbb{Q}_{Z}\boxtimes
H_{c}^{\ast}(F)[2d]).$

We prove the lemma by constructing a global section of (4.27) restricting to
isomorphisms on trivializable open subsets.

Let $U\subseteq Y$ be connected such that
$\zeta|_{\zeta^{-1}(U)}\operatorname{\cong}\operatorname{pr}_{1}$ is
projection from $U\times Y$. We have

(4.28) $\mathcal{H}om(\mathbb{Q}_{U}\boxtimes
H_{c}^{\ast}(F),\mathbb{Q}_{U}\boxtimes
H_{c}^{\ast}(F))\operatorname{\cong}\zeta_{\ast}\mathcal{H}om(\mathbb{Q}_{U\times
F},\mathbb{Q}_{U\times F}\boxtimes H^{\ast}_{c}(F)[2d])$

since $\zeta_{!}\mathbb{Q}_{U\times
Y}\operatorname{\cong}\mathbb{Q}_{U}\boxtimes H_{c}^{\ast}(F)$ by the Künneth
formula. Global sections in the RHS of (4.28) are given by

(4.29) $\begin{split}Hom_{D(U\times F)}(\mathbb{Q}_{U\times F},\bigoplus_{n\in
N}\mathbb{Q}_{U\times F}[2d-n])&=\bigoplus_{n\in N}H^{2d-n}(U\times F)\\\
&\supseteq\bigoplus_{n\in N}H^{0}(U)\otimes H^{2d-n}(F)\end{split}$

where $N$ is the graded dimension of $H_{c}^{\ast}(F)$. The identity element
in the LHS of (4.28) corresponds to a generic element in RHS of (4.29), one
coordinate for each element of $N$.

The RHS of (4.27) has global sections equal to

(4.30) $\operatorname{Hom}_{D(Z)}(\mathbb{Q}_{Z},\bigoplus_{n\in
N}\mathbb{Q}_{Z}[2d-n])=\bigoplus_{n\in N}H^{2d-n}(Z).$

For every $n\in N$ we have by assumption an element $f_{n}\in H^{2d-n}(Z)$
restricting to a basis of $H^{2d-n}(F)$, where we have
$H^{2d-n}(F)=H^{n}_{c}(F)$ by Poincaré duality; i.e., $(f_{n})_{n\in N}$
restricts to a basis of $H^{\ast}(F)\operatorname{\cong}H_{c}^{\ast}(F)$. The
element $(f_{n})_{n\in N}$ corresponds on the LHS of (4.27) to our desired
isomorphism in
$\operatorname{Hom}_{D(Y)}(\zeta_{!}\zeta^{\ast}\mathbb{Q}_{Y},\mathbb{Q}_{Y}\boxtimes
H_{c}^{\ast}(F))$ since it restricts to isomorphisms over trivializable open
sets by Lemma 4.2.3. ∎

######  Remark 4.2.5.

We apply Lemma 4.2.4 directly to the equivariant derived category by
considering the finite dimensional smooth approximations in our smooth model
(as in, e.g., [AF21, Lem. 2.1]). For example, the map

(4.31) $ET\times_{T}G/B\to
ET\times_{T}G/P_{s},\qquad[e,x]\mapsto[e,\pi_{s}(x)]$

satisfies the Leray-Hirsch theorem.

We are now ready to begin our case-by-case analysis of root types. We align
our numeration and terminology with Definition 3.2.1.

#### (a) Compact imaginary

######  Example 4.2.6 ($G_{\mathbb{R}}$ maximal compact subgroup of $G$).

Let $G$ be any connected complex reductive algebraic group and $\theta$ be the
identity involution. Then $X$ is a homogeneous space for $K=G$, and the simple
reflection $s$ is compact imaginary (a) with respect to the single
Lusztig–Vogan parameter $(X,\mathbb{Q}_{X})$. We have

(4.32)
$\pi_{s}^{\ast}\pi_{s\ast}\mathbb{Q}_{X}=\mathbb{Q}_{X}\oplus\mathbb{Q}_{X}[-2]$

by Lemma 4.2.4. It follows that

(4.33) $m(\pi_{s}^{\ast}\pi_{s\ast}\mathbb{Q}_{X})={\mathbb{Q}_{X}}(T_{s}+1)$

by (4.8).

In general, for a compact imaginary reflection $s$ with respect to
$(\mathcal{O},\mathcal{L})$, the maps in (4.12) have the form

(4.34)
${K/K_{[x,s]}}$${K/K_{x}}$${K/K_{x}}$$\scriptstyle{\mu}$$\scriptstyle{\pi}$

for $x\in\mathcal{O}=KxB/B$. In this case $\mu$ is smooth with fiber the
affine line by (4.13), and $\pi=\mu$ up to a $K$-equivariant automorphism of
$K/K_{[x,s]}=KxB\times_{B}BsB/B$. In particular, $\pi^{\ast}\mathcal{F}$ and
$\mu^{\ast}\mathcal{F}$ are isomorphic in $D_{K}^{b}(K/K_{[x,s]})$. We have

(4.35) $\mu_{!}\pi^{\ast}\mathcal{L}=\mathcal{L}[-2]$

by Lemma 4.2.4. It follows that

(4.36)
$\begin{split}m(k_{\mathcal{O}!}\mu_{!}\pi^{\ast}\mathcal{L})&=q\mathcal{L}\\\
&=\mathcal{L}T_{s}\end{split}$

by (4.8), so (4.11) and (4.14) hold true.

#### (b) Complex

######  Example 4.2.7.

$(G_{\mathbb{R}}=SL_{2}(\mathbb{C}))$ Set $G=SL_{2}(\mathbb{C})\times
SL_{2}(\mathbb{C})$ and let $\theta$ be the involution that switches the
factors; i.e., $\theta(g_{1},g_{2})=(g_{2},g_{1})$. Then $K\simeq
SL_{2}(\mathbb{C})$ is the diagonal subgroup in $SL_{2}(\mathbb{C})\times
SL_{2}(\mathbb{C})$, and the flag variety is isomorphic to
$\mathbb{P}^{1}\times\mathbb{P}^{1}$. Let $B_{K}$ be the Borel subgroup of
upper triangular matrices in $SL_{2}(\mathbb{C})$. We can identify
$D^{b}_{SL_{2}(\mathbb{C})}(\mathbb{P}^{1}\times\mathbb{P}^{1})=D^{b}_{B_{K}}(\mathbb{P}^{1})$
via induction and study $B_{K}$-equivariant local systems on $\mathbb{P}^{1}$.
The $B_{K}$-orbits on $\mathbb{P}^{1}$ are

$\mathcal{Q}_{0}:=\\{[1,0]\\}\text{ and
}\mathcal{O}:=\\{[x,1]\\}\simeq\mathbb{C}.$

The simple reflection $s$ is complex ascent (b1) with respect to the
Lustig–Vogan parameter $(\mathcal{Q}_{0},\mathbb{Q}_{\mathcal{Q}_{0}})$ and
complex decent (b2) with respect to the parameter
$(\mathcal{O},\mathbb{Q}_{\mathcal{O}})$.

Let $i_{0}:\mathcal{Q}_{0}\rightarrow\mathbb{P}^{1}$ and
$j:\mathcal{O}\rightarrow\mathbb{P}^{1}$ be the closed and open embeddings.
Because $i_{0!}=i_{0*}$, we have

(4.37) $\displaystyle\pi_{s}^{*}\pi_{s*}i_{0!}\mathbb{Q}_{\mathcal{Q}_{0}}$
$\displaystyle=\pi_{s}^{*}\pi_{s*}i_{0*}\mathbb{Q}_{\mathcal{Q}_{0}}$
$\displaystyle=\pi_{s}^{*}\mathbb{Q}_{\mathrm{pt}}$
$\displaystyle=\mathbb{Q}_{\mathbb{P}^{1}}.$

By applying Lemma 4.2.4 to the smooth map $\pi_{s}\circ j$, we compute

(4.38) $\displaystyle\pi_{s}^{*}\pi_{s*}j_{!}\mathbb{Q}_{\mathcal{O}}$
$\displaystyle=\pi_{s}^{*}(\pi_{s}\circ j)_{!}\mathbb{Q}_{\mathcal{O}}$
$\displaystyle=\pi_{s}^{*}(\pi_{s}\circ j)_{!}(\pi_{s}\circ
j)^{*}\mathbb{Q}_{\mathrm{pt}}$
$\displaystyle=\pi_{s}^{*}\mathbb{Q}_{\mathrm{pt}}[-2]$
$\displaystyle=\mathbb{Q}_{\mathbb{P}^{1}}[-2].$

It follows from (4.8) that (4.11) holds true for the above complex
reflections.

In general if $s$ is a complex ascent with respect to
$(\mathcal{O}=KxB/B,\mathcal{L})$ we have

(4.39)
${KxB\times_{B}BsB/B}$${KyB/B}$${KxB/B}$$\scriptstyle{\mu}$$\scriptstyle{\pi}$

where $\mu$ is an isomorphism. For
$\mathcal{L}=k_{\mathcal{O}}^{\ast}k_{\mathcal{O}!}\mathcal{L}$, we have

(4.40)
$\mu_{!}\pi^{\ast}\mathcal{L}=\widehat{\mathcal{L}}_{\widehat{\mathcal{O}}\smallsetminus\mathcal{O}}$

where $\widehat{\mathcal{O}}=\bigcup_{x\in\mathcal{O}}L_{x}^{s}$ (as in
Definition 3.2.1) and $\widehat{\mathcal{L}}$ is the extension of
$\mathcal{L}$ to $\widehat{\mathcal{O}}$, so (4.14) holds true.

If $s$ is a complex descent with respect to $(\mathcal{O},\mathcal{L})$, let
$y$ be such that $\widehat{\mathcal{O}}=KyB/B\cup KxB/B$ and denote by
$k_{\widehat{\mathcal{O}}}$ the inclusion of $\widehat{\mathcal{O}}$ in $G/B$.
Then $s$ is a complex ascent with respect to
$(\mathcal{O}^{\prime}=KyB/B,\mathcal{L}^{\prime})$, where
$(\mathcal{O}^{\prime},\mathcal{L}^{\prime})$ is the corresponding element in
$\mathscr{D}$ under the identification $K_{y}/K_{y}^{0}=K_{x}/K_{x}^{0}$. The
cartesian diagram

(4.41)
${KyB\times_{B}P_{s}/B}$${\widehat{\mathcal{O}}}$${KyB/B}$${KyP_{s}/P_{s}}$$\scriptstyle{\bar{\mu}}$$\scriptstyle{\bar{\pi}}$$\scriptstyle{\pi_{s}k_{\widehat{\mathcal{O}}}}$$\scriptstyle{\pi_{s}^{\prime}}$

gives the unique extension of $\mathcal{L}$ to $\widehat{\mathcal{O}}$,
$\widehat{\mathcal{L}}:=\bar{\mu}_{\ast}\bar{\pi}^{\ast}\mathcal{L}^{\prime}$,
since $\bar{\mu}$ is an isomorphism. By change of base, the $K$-equivariant
local system $\widehat{\mathcal{L}}$ is the pull-back of a $K$-equivariant
local system on $KyP_{s}/P_{s}$, so

(4.42)
$m(\pi_{s}^{\ast}\pi_{s\ast}(k_{\widehat{\mathcal{O}}!}\widehat{\mathcal{L}}))=(1+q)\widehat{\mathcal{L}}$

by Lemma 4.2.4 and (4.8). The distinguished triangle

(4.43)
$\pi_{s}^{\ast}\pi_{s\ast}k_{\mathcal{O}!}\mathcal{L}\to\pi_{s}^{\ast}\pi_{s\ast}k_{\widehat{\mathcal{O}}!}\to\pi_{s}^{\ast}\pi_{s\ast}k_{\mathcal{O}^{\prime}!}\mathcal{L}^{\prime}\overset{+1}{\longrightarrow}$

is obtained by applying the triangulated functor
$\pi_{s}^{\ast}\pi_{s\ast}k_{\widehat{\mathcal{O}}!}$ to the open-closed
distinguished triangle of
$\widehat{\mathcal{O}}=\mathcal{O}\cup\mathcal{O}^{\prime}$, and satisfies
(4.10) since $\pi_{s}k_{\mathcal{O}^{\prime}}$ is finite. By the complex
ascent case (4.40), we have

(4.44)
$\begin{split}m(\pi_{s}^{\ast}\pi_{s\ast}(k_{\mathcal{O}!}\mathcal{L}))&=m(\pi_{s}^{\ast}\pi_{s\ast}k_{\widehat{\mathcal{O}}!})-m(\pi_{s}^{\ast}\pi_{s\ast}k_{\mathcal{O}^{\prime}!}\mathcal{L}^{\prime})\\\
&=q\widehat{\mathcal{L}}\\\ &=\mathcal{L}(T_{s}+1)\end{split}$

by Definition 3.2.1. We conclude that (4.11) and (4.14) hold true for all
complex reflections.

#### (c) Type II

######  Example 4.2.8.

$(G_{\mathbb{R}}=PSL_{2}(\mathbb{R}))$ Let $G=PSL_{2}(\mathbb{C})$, and take
$\theta$ to be the involution

$\theta(g)=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}g\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$

Then

$K=\left\\{\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix}\mid
a\in\mathbb{C}^{\times}\right\\}\sqcup\left\\{\begin{pmatrix}0&b\\\
b^{-1}&0\end{pmatrix}\mid b\in\mathbb{C}^{\times}\right\\}\hskip
5.69054pt(\mathrm{mod}\pm I)$

is disconnected and isomorphic to the complex orthogonal group. The group $K$
acts on the flag variety $X=\mathbb{P}^{1}$ with two orbits,

$\mathcal{Q}_{0,\infty}=\\{[0,1],[1,0]\\}\text{ and }\mathcal{O}=\\{[x,y]\mid
x\neq 0,y\neq 0\\}\simeq\mathbb{C}^{\times}.$

The stabilizer $K_{[1,0]}=K_{[0,1]}\simeq\mathbb{C}^{\times}$ is connected, so
there is one irreducible $K$-equivariant local system on the closed orbit
$\mathcal{Q}_{0,\infty}$. For $[x,y]\in\mathcal{O}$, the stabilizer
$K_{[x,y]}\simeq\mathbb{Z}/2\mathbb{Z}$ is disconnected, so there are two
irreducible $K$-equivariant local systems on the open orbit $\mathcal{O}$,
corresponding to the trivial and sign representations of
$\mathbb{Z}/2\mathbb{Z}$. Denote the non-trivial $K$-equivariant local system
by $\mathcal{L}$. The reflection $s$ is noncompact imaginary of type II (c1)
with respect to $(\mathcal{Q}_{0,\infty},\mathbb{Q}_{\mathcal{Q}_{0,\infty}})$
and real of type II (c2) with respect to
$(\mathcal{O},\mathbb{Q}_{\mathcal{O}})$ and $(\mathcal{O},\mathcal{L})$.

The irreducible $K$-equivariant local systems on $\mathcal{O}$ have the same
underlying local system and only differ in their $K$-equivariant structures.
This can be seen by examining the long exact sequence in homotopy groups
associated to the fibration

$\mathbb{Z}/2\mathbb{Z}\simeq K_{[x,y]}\hookrightarrow K\twoheadrightarrow
K/K_{[x,y]}\simeq\mathcal{O}.$

Indeed, the long exact sequence in homotopy groups has the form

$\cdots\rightarrow\pi_{1}(\mathcal{O})\xrightarrow{\alpha}\pi_{0}(K_{[x,y]})\xrightarrow{\beta}\pi_{0}(K)\rightarrow
1,$

so by exactness, $\beta$ is an isomorphism and the image of $\alpha$ is $1$.
Pulling back via $\alpha$ gives a functor

$\pi_{0}(K_{[x,y]})\mathrm{-mod}\xrightarrow{\alpha^{*}}\pi_{1}(\mathcal{O})\mathrm{-mod},$

which determines the underlying local system of a $K$-equivariant local system
on $\mathcal{O}$. Because $\alpha$ is trivial, we conclude that the
$K$-equivariant local systems $\mathbb{Q}_{\mathcal{O}}$ and $\mathcal{L}$
must have the same (trivial) underlying local system.

Let $i_{0,\infty}:\mathcal{Q}_{0,\infty}\rightarrow X$ be inclusion, so
$\pi_{s}\circ i_{0,\infty}$ is the projection of two points onto one point.
There are two irreducible $K$-equivariant local systems on a point,
$\mathbb{Q}_{\mathrm{pt}}$ and $\mathcal{L}_{\mathrm{sgn}}$, corresponding to
the two irreducible representations of the component group of $K$, which is
isomorphic to $\mathbb{Z}/2\mathbb{Z}$. Under the equivalence between
$K$-equivariant local systems on a point and
$\mathbb{Z}/2\mathbb{Z}$-representations, $(\pi_{s}\circ
i_{0,\infty})_{*}\mathbb{Q}_{\mathcal{Q}_{0,\infty}}$ corresponds to the
regular representation, so

$(\pi_{s}\circ
i_{0,\infty})_{*}\mathbb{Q}_{\mathcal{Q}_{0,\infty}}=\mathbb{Q}_{\mathrm{pt}}\oplus\mathcal{L}_{\mathrm{sgn}}.$

Hence

(4.45)
$\displaystyle\pi_{s}^{*}\pi_{s*}i_{0,\infty!}\mathbb{Q}_{\mathcal{Q}_{0,\infty}}$
$\displaystyle=\pi_{s}^{*}(\mathbb{Q}_{\mathrm{pt}}\oplus\mathcal{L}_{\mathrm{sgn}})$
$\displaystyle=\mathbb{Q}_{X}\oplus\widehat{\mathcal{L}},$

where $\widehat{L}=\pi_{s}^{*}\mathcal{L}_{\mathrm{sgn}}$ is the unique
extension of $\mathcal{L}$ to $X$. This shows that (4.11) holds for
$(\mathcal{Q}_{0,\infty},\mathbb{Q}_{\mathcal{Q}_{0,\infty}})$.

Let $j:\mathcal{O}\rightarrow X$ be inclusion of the open orbit. First, note
that

(4.46)
$m(\pi_{s}^{*}\pi_{s*}\mathbb{Q}_{X})=m(\pi_{s}^{*}\pi_{s*}i_{0,\infty!}\mathbb{Q}_{\mathcal{Q}_{0,\infty}})+m(\pi_{s}^{*}\pi_{s*}j_{!}\mathbb{Q}_{\mathcal{O}})$

by (4.10). (We obtain a distinguished triangle satisfying the conditions of
(4.10) by applying the triangulated functor $\pi_{s}^{*}\pi_{s*}$ to the open-
closed distinguished triangle for the pair $(j,i_{0,\infty})$ and using
(4.45).) We have

$\pi_{s}^{*}\pi_{s*}\mathbb{Q}_{X}=\mathbb{Q}_{X}\oplus\mathbb{Q}_{X}[-2]$

by Lemma 4.2.4, so

(4.47)
$m(\pi_{s}^{*}\pi_{s*}\mathbb{Q}_{X})=(1+q)(\mathbb{Q}_{\mathcal{Q}_{0,\infty}}+\mathbb{Q}_{\mathcal{O}}).$

Combining equations (4.46), (4.47) and (4.45), we see that

$m(\pi_{s}^{*}\pi_{s*}j_{!}\mathbb{Q}_{\mathcal{O}})=q\mathbb{Q}_{\mathcal{O}}+(q-1)\mathbb{Q}_{\mathcal{Q}_{0,\infty}}-\mathcal{L},$

so (4.11) holds for the parameter $(\mathcal{O},\mathbb{Q}_{\mathcal{O}})$.

Finally, to see that (4.11) holds for the parameter
$(\mathcal{O},\mathcal{L})$, we apply Lemma 4.2.4 to $\pi_{s}$ and
$\mathcal{L}_{\mathrm{sgn}}$. Specifically, if we denote the extension of
$\mathcal{L}$ to $X$ by $\widehat{\mathcal{L}}$ as above, then
$\pi_{s}^{*}\mathcal{L}_{\mathrm{sgn}}=\widehat{\mathcal{L}}$, so

(4.48)
$\pi_{s}^{*}\pi_{s*}\widehat{\mathcal{L}}=\widehat{\mathcal{L}}\oplus\widehat{\mathcal{L}}[-2].$

Analogously to the previous case, we have

(4.49)
$m(\pi_{s}^{*}\pi_{s*}\widehat{\mathcal{L}})=m(\pi_{s}^{*}\pi_{s*}i_{0,\infty!}\mathbb{Q}_{\mathcal{Q}_{0,\infty}})+m(\pi_{s}^{*}\pi_{s*}j_{!}\mathcal{L}),$

so $m(\pi_{s}^{*}\pi_{s*}j_{!}\mathcal{L})$ can be computed using (4.48) and
(4.45). The result of this computation is that (4.11) also holds for the
parameter $(\mathcal{O},\mathcal{L})$, so it holds for all type II reflections
in this example.

In general, for a noncompact imaginary reflection $s$ of type II with respect
to $(\mathcal{O}=KxB/B,\mathcal{L})$, we have

(4.50)
${KxB\times_{B}P_{s}/B}$${\widehat{\mathcal{O}}}$${KxB/B}$${KxP_{s}/P_{s}}$$\scriptstyle{\bar{\mu}}$$\scriptstyle{\bar{\pi}}$$\scriptstyle{\pi_{s}k_{\widehat{\mathcal{O}}}}$$\scriptstyle{\pi_{s}^{\prime}}$

such that $\bar{\mu}$ and $\pi_{s}^{\prime}$ are finte maps of degree two. We
have

(4.51)
$\bar{\mu}_{!}\bar{\pi}^{\ast}\mathcal{L}=(\pi_{s}k_{\widehat{\mathcal{O}}})^{\ast}\operatorname{Ind}_{K_{x}}^{K\cap
xP_{s}x^{-1}}(\mathcal{L})=\widehat{\mathcal{L}}_{1}\oplus\widehat{\mathcal{L}}_{2}$

for local systems $\widehat{\mathcal{L}}_{1}$ and $\widehat{\mathcal{L}}_{2}$
on $\widehat{\mathcal{O}}$, which are distinct when restricted to the open
orbit $\mathcal{O}^{\prime}=K/K_{y}$ in $\widehat{\mathcal{O}}$.

By Definition 3.2.1,

(4.52)
$\begin{split}m(\pi_{s}^{\ast}\pi_{s\ast}k_{\mathcal{O}!}\mathcal{L})&=m(\widehat{\mathcal{L}}_{1})+m(\widehat{\mathcal{L}}_{2})\\\
&=2\mathcal{L}+(\widehat{\mathcal{L}}_{1}+\widehat{\mathcal{L}}_{2})|_{\mathcal{O}^{\prime}}\\\
&=\mathcal{L}(T_{s}+1)\end{split}$

so (4.11) holds true for all type II noncompact imaginary reflections.

Suppose $s$ is type II real with respect to $(\mathcal{O},\mathcal{L})$, and
let $y$ be such that $\widehat{\mathcal{O}}=KyB/B\cup KxB/B$. By assumption,
there exists $\mathcal{L}^{\prime}$ on $\mathcal{O}^{\prime}=KyB/B$ such that

(4.53)
${KyB\times_{B}P_{s}/B}$${\widehat{\mathcal{O}}}$${KyB/B}$${KyP_{s}/P_{s}}$$\scriptstyle{\bar{\mu}}$$\scriptstyle{\bar{\pi}}$$\scriptstyle{\pi_{s}k_{\widehat{\mathcal{O}}}}$$\scriptstyle{\pi_{s}^{\prime}}$

gives rise to one of the local systems $\widehat{\mathcal{L}}$ from (4.51)
satisfying $\widehat{\mathcal{L}}|_{\mathcal{O}}=\mathcal{L}$. Then $s$ is
type II noncompact imaginary with respect to
$(\mathcal{O}^{\prime},\mathcal{L}^{\prime})$. By (4.51),
$\widehat{\mathcal{L}}$ is the pull-back of a $K$-equivariant local system on
$KyP_{s}/P_{s}$, so

(4.54)
$m(\pi_{s}^{\ast}\pi_{s\ast}(k_{\widehat{\mathcal{O}}!}\widehat{\mathcal{L}}))=(q+1)\widehat{\mathcal{L}}$

by Lemma 4.2.4 and (4.8). By the distinguished triangle (4.43) corresponding
to the open-closed decomposition
$\widehat{\mathcal{O}}=\mathcal{O}\cup\mathcal{O}^{\prime}$ and the type II
noncompact imaginary case (4.52), we have

(4.55)
$\begin{split}m(\pi_{s}^{\ast}\pi_{s\ast}(k_{\mathcal{O}!}\mathcal{L}))&=m(\pi_{s}^{\ast}\pi_{s\ast}k_{\widehat{\mathcal{O}}!}\widehat{\mathcal{L}})-(2\mathcal{L}^{\prime}+\mathcal{L}+\widehat{\mathcal{L}}_{2}|_{\mathcal{O}})\\\
&=q\mathcal{L}-\widehat{\mathcal{L}}_{2}|_{\mathcal{O}}+(q-1)\mathcal{L}^{\prime}\\\
&=\mathcal{L}(T_{s}+1)\end{split}$

where $\widehat{\mathcal{L}}_{2}$ is the other $K$-equivariant local system on
$\widehat{\mathcal{O}}$. It follows that (4.11) and (4.14) hold true for every
type II real reflection.

#### (d) Type I

######  Example 4.2.9.

$(G_{\mathbb{R}}=SL_{2}(\mathbb{R}))$ Let $G,K$, and $\theta$ be as in Example
3.2.4. Denote by $i_{0}$ and $i_{\infty}$ the inclusions of the single-point
orbits $\mathcal{Q}_{0}$ and $\mathcal{Q}_{\infty}$ into $X=\mathbb{P}^{1}$,
respectively, and $j:\mathcal{O}\rightarrow X$ the inclusion of the open
orbit. The reflection $s$ is type I noncompact imaginary (d1) for
$(\mathcal{Q}_{0},\mathbb{Q}_{\mathcal{Q}_{0}})$ and
$(\mathcal{Q}_{\infty},\mathbb{Q}_{\mathcal{Q}_{\infty}})$, and type I real
(d2) for $(\mathcal{O},\mathbb{Q}_{\mathcal{O}})$.

As in equation (4.37), we have

(4.56)
$\begin{split}\pi_{s}^{\ast}\pi_{s\ast}i_{0!}\mathbb{Q}_{\mathcal{Q}_{0}}&=\mathbb{Q}_{X},\text{
and }\\\
\pi_{s}^{\ast}\pi_{s\ast}i_{\infty!}\mathbb{Q}_{\mathcal{Q}_{\infty}}&=\mathbb{Q}_{X}.\\\
\end{split}$

It follows that (4.11) holds for
$(\mathcal{Q}_{0},\mathbb{Q}_{\mathcal{Q}_{0}})$ and
$(\mathcal{Q}_{\infty},\mathbb{Q}_{\mathcal{Q}_{\infty}})$.

To show that (4.11) holds for the parameter
$(\mathcal{O},\mathbb{Q}_{\mathcal{O}})$, we proceed as in Example 4.2.8. Let
$i_{0,\infty}:\mathcal{Q}_{0}\cup\mathcal{Q}_{\infty}\rightarrow X$ be the
inclusion of the union of the two single point orbits into $X$. By applying
the triangulated functor $\pi_{s}^{*}\pi_{s*}$ to the open-closed
distinguished triangle for the pair $(i_{0,\infty},j)$, we obtain the
distinguished triangle

(4.57)
$\pi_{s}^{*}\pi_{s*}j_{!}\mathbb{Q}_{\mathcal{O}}\rightarrow\pi_{s}^{*}\pi_{s*}\mathbb{Q}_{X}\rightarrow\pi_{s}^{*}\pi_{s*}i_{0,\infty*}\mathbb{Q}_{\mathcal{Q}_{0}\cup\mathcal{Q}_{\infty}}\xrightarrow{+1}.$

By applying (4.10) to the triangle (4.57), we see that

(4.58)
$m(\pi_{s}^{*}\pi_{s*}\mathbb{Q}_{X})=m(\pi_{s}^{*}\pi_{s*}j_{!}\mathbb{Q}_{\mathcal{O}})+m(i_{0,\infty*}\mathbb{Q}_{\mathcal{Q}_{0}\cup\mathcal{Q}_{\infty}}).$

By Lemma 4.2.4,

(4.59)
$\pi_{s}^{*}\pi_{s*}\mathbb{Q}_{X}=\mathbb{Q}_{X}\oplus\mathbb{Q}_{X}[-2],$

and

(4.60)
$\pi_{s}^{*}\pi_{s*}i_{0,\infty*}\mathbb{Q}_{\mathcal{Q}_{0}\cup\mathcal{Q}_{\infty}}=\mathbb{Q}_{X}\oplus\mathbb{Q}_{X}.$

Combining (4.58), (4.59), and (4.60), we have

$m(\pi_{s}^{*}\pi_{s*}j_{!}\mathbb{Q}_{\mathcal{O}})=(q-1)(\mathbb{Q}_{\mathcal{Q}_{0}}+\mathbb{Q}_{\mathcal{Q}_{\infty}}+\mathbb{Q}_{\mathcal{O}}).$

It follows that (4.11) holds for $(\mathcal{O},\mathbb{Q}_{\mathcal{O}})$.

In general, let $s$ be noncompact imaginary reflection of type I with respect
to $(KxB/B,\mathcal{L})$. We have

(4.61)
${KxB\times_{B}P_{s}/B}$${\widehat{\mathcal{O}}}$${KxB/B}$${KxP_{s}/P_{s}}$$\scriptstyle{\bar{\mu}}$$\scriptstyle{\bar{\pi}}$$\scriptstyle{\pi_{s}k_{\widehat{\mathcal{O}}}}$$\scriptstyle{\pi_{s}^{\prime}}$

such that $\bar{\mu}$ and $\pi_{s}^{\prime}$ are isomorphisms, so

(4.62)
$\widehat{\mathcal{L}}=\bar{\mu}_{!}\bar{\pi}^{\ast}\mathcal{L}=(\pi_{s}k_{\widehat{\mathcal{O}}})^{\ast}\pi_{s!}^{\prime}\mathcal{L}$

extends $\mathcal{L}$ to $\widehat{\mathcal{O}}$.

By Definition 3.2.1, we have

(4.63)
$\begin{split}m(\pi_{s}^{\ast}\pi_{s\ast}k_{\mathcal{O}!}\mathcal{L})&=m(\widehat{\mathcal{L}})\\\
&=\mathcal{L}+\widehat{\mathcal{L}}|_{\mathcal{O}^{\prime}}+\widehat{\mathcal{L}}|_{\mathcal{O}^{\prime\prime}}\\\
&=\mathcal{L}(T_{s}+1)\end{split}$

where $\mathcal{O}^{\prime}$ and $\mathcal{O}^{\prime\prime}$ are the
$K$-orbits such that
$\widehat{\mathcal{O}}=\mathcal{O}\cup\mathcal{O}^{\prime}\cup\mathcal{O}^{\prime\prime}$
and
$\dim(\mathcal{O})=\dim(\mathcal{O}^{\prime\prime})=\dim(\mathcal{O}^{\prime})-1$.
By (4.20), (4.11) holds true for all noncompact imaginary reflections of type
I.

Now assume $s$ is real type I with respect to $(\mathcal{O},\mathcal{L})$, so
there exists $\mathcal{L}^{\prime}$ on $\mathcal{O}^{\prime}=KyB/B$ such that

(4.64)
${KyB\times_{B}P_{s}/B}$${\widehat{\mathcal{O}}}$${KyB/B}$${KyP_{s}/P_{s}}$$\scriptstyle{\bar{\mu}}$$\scriptstyle{\bar{\pi}}$$\scriptstyle{\pi_{s}k_{\widehat{\mathcal{O}}}}$$\scriptstyle{\pi_{s}^{\prime}}$

gives rise to the unique local system $\widehat{\mathcal{L}}$ from (4.62). We
have

(4.65)
$m(\pi_{s}^{\ast}\pi_{s\ast}(k_{\widehat{\mathcal{O}}!}\widehat{\mathcal{L}}))=(q+1)\widehat{\mathcal{L}}$

by Lemma 4.2.4 and (4.8). By the distinguished triangle (4.43) corresponding
to the open-closed decomposition
$\widehat{\mathcal{O}}=\mathcal{O}\cup(\mathcal{O}^{\prime}\cup\mathcal{O}^{\prime\prime})$
and the type I noncompact imaginary case (4.63), we have

(4.66)
$\begin{split}m(\pi_{s}^{\ast}\pi_{s\ast}(k_{\mathcal{O}!}\mathcal{L}))&=m(\pi_{s}^{\ast}\pi_{s\ast}(k_{\widehat{\mathcal{O}}!}\widehat{\mathcal{L}}))-2m(k_{\widehat{\mathcal{O}}!}\widehat{\mathcal{L}})\\\
&=(q-1)\widehat{\mathcal{L}}\\\ &=\mathcal{L}(T_{s}+1)\end{split}$

by Definition 3.2.1. It follows that (4.11) and (4.14) hold true for every
type I real reflection.

#### (e) Real Nonparity

######  Example 4.2.10.

$(G_{\mathbb{R}}=SL_{2}(\mathbb{R}))$ We return to the setting of Examples
3.2.4 and 4.2.9. With respect to the Möbius band local system $\mathcal{L}$ on
the open orbit $\mathcal{O}$, the reflection $s$ is type (e), real nonparity.
In particular, the local system $\mathcal{L}$ does not extend (as a local
system) to $X$, so we have

$j_{!}\mathcal{L}=j_{*}\mathcal{L},$

where $j:\mathcal{O}\rightarrow X$ is inclusion. Hence we can compute
$\pi_{s*}j_{!}\mathcal{L}=(\pi_{s}\circ j)_{*}\mathcal{L}$ by computing the
sheaf cohomology of $\mathbb{C}^{\times}$ with coefficients in the local
system $\mathcal{L}$. We will show that this cohomology is zero.

Forgetting equivariance, the local system $\operatorname{For}\mathcal{L}$
consists of the data of a one-dimensional $\mathbb{Q}$-vector space $V$ and
the monodromy map $m=-1:V\rightarrow V$. The cohomology of
$\mathbb{C}^{\times}$ with coefficients in $\operatorname{For}\mathcal{L}$ is
given by the invariants and coinvariants of $m$ [RW21, Ex. 17.6]:

$H^{i}(\mathbb{C}^{\times};\operatorname{For}\mathcal{L})=\begin{cases}V^{m}&i=0;\\\
V_{m}&i=1.\end{cases}$

Since $V^{m}=V_{m}=0$, we see that on the non-equivariant level,
$\pi_{s*}j_{!}\operatorname{For}\mathcal{L}=0$. Hence the equivariant complex
$\pi_{s*}j_{!}\mathcal{L}$ must also vanish.

Alternately, we can see this using the finite Galois covering

$f:\mathbb{C}^{\times}\rightarrow\mathbb{C}^{\times};z\mapsto z^{2}.$

Again, we forget equivariance. In the equivalence between local systems and
representations of fundamental groups, pushing along a finite covering map
corresponds to inducing representations. Hence the local system
$f_{*}\mathbb{Q}_{\mathbb{C}^{\times}}$ corresponds to the induced
representation $\operatorname{Ind}_{2\mathbb{Z}}^{\mathbb{Z}}\mathrm{triv}$,
which is isomorphic to the regular representation of $\mathbb{Z}/2\mathbb{Z}$.
We conclude that

$f_{*}\mathbb{Q}_{\mathbb{C}^{\times}}=\mathbb{Q}_{\mathbb{C}^{\times}}\oplus\operatorname{For}\mathcal{L}.$

Pushing along $\pi_{s}\circ j$, we see that

$\pi_{s*}j_{*}\mathbb{Q}_{\mathbb{C}^{\times}}=\pi_{s*}j_{*}f_{*}\mathbb{Q}_{\mathbb{C}^{\times}}=\pi_{s*}j_{*}\mathbb{Q}_{\mathbb{C}^{\times}}\oplus\pi_{s*}j_{*}\operatorname{For}\mathcal{L},$

because $\pi_{s}\circ j=\pi_{s}\circ j\circ f$. Hence
$\pi_{s*}j_{!}\operatorname{For}\mathcal{L}=0$.

From either of these two computations, we conclude that

(4.67) $\pi_{s}^{*}\pi_{s*}j_{!}\mathcal{L}=0.$

It follows that (4.11) holds for this type (e) reflection.

In general, suppose $s$ is real nonparity with respect to
$(\mathcal{O},\mathcal{L})$. We start with a Lemma.

######  Lemma 4.2.11.

For $x\in\mathcal{O}$,

(4.68) $\widehat{\mathcal{O}}=K\times_{K\cap xP_{s}x^{-1}}L_{x}^{s}.$

###### Proof.

The quotient in (4.68) is an algebraic variety since $L:=L_{x}^{s}$ is
quasiprojective and $K\cap xP_{s}x^{-1}$ is a closed subgroup of $K$. Define

(4.69) $\eta\,\colon K\times_{K\cap xP_{s}x^{-1}}xP_{s}/B\to KxP_{s}/B$

by multiplication. We have

(4.70) $\eta^{-1}(kxpB/B)=\left\\{[k^{\prime},xp^{\prime}]\mid
k^{\prime}xp^{\prime}B/B=kxpB/B\right\\}=\left\\{[k,xp]\right\\}$

since $[k^{\prime},xp^{\prime}]=[k,k^{-1}k^{\prime}xp^{\prime}]=[k,xp]$ by a
routine calculation. It follows that $\eta$ is an isomorphism since
$KxP_{s}/B$ is smooth and $\eta$ is bijective. ∎

In particular, (4.68) shows that $K\times_{K\cap xP_{s}x^{-1}}L$ is a Zariski
locally trivial fiber bundle over $KxP_{s}/P_{s}=K/(K\cap xP_{s}x^{-1})$ with
fiber $L$ (even when $K\cap xP_{s}x^{-1}$ need not be a parabolic subgroup of
$K$). There are finitely many $K$-orbits in $\widehat{\mathcal{O}}$, which
correspond to $K\cap xP_{s}x^{-1}$ orbits in $L$ via (4.68); this is computed
explicitly in [Vog82, Lem. 5.1].

We have an equivalence of t-categories

(4.71) $D_{K}^{b}(KxP_{s}/B)=D^{b}_{K\cap xP_{s}x^{-1}}(L)$

by induction. If $y\in KxP_{s}/B$ and $k\,\colon KyB/B\to KxP_{s}/B$ is
inclusion, then $\mathcal{L}=k^{\ast}k_{!}\mathcal{L}$ corresponds to the same
representation of $K_{y}/K_{y}^{0}$ in either category. Under the equivalence
(4.71),

(4.72) $\pi_{s_{\ast}}\mathcal{F}=p_{\ast}\mathcal{F}$

where $p\,\colon L\to\operatorname{pt}$ maps to a point.

It follows from Example 4.2.10 that for any real nonparity reflection we have

(4.73) $\mathcal{L}(T_{s}+1)=0.$

This completes the proof of Lemma 4.2.1.

###  4.3. Proof of Theorem 4.1.3

For $y\in G/B$, let $k^{\prime}\,\colon KyP_{s}/B\to G/B$. Let $U$ be the
union of the open $K$-orbit $Ky_{0}B/B$ in $KyP_{s}/B$ and the complement of
$\overline{KyP_{s}/B}$ in $G/B$. Let $j\,\colon U\to G/B$ be the corresponding
open embedding. Let $Y$ be the complement of $U$ in $G/B$, so $Y$ is the
closure in $G/B$ of a (possibly empty) disjoint union of closed $K$-orbits
$Ky_{i}B/B$ in $KyP_{s}/B$. Let $i\,\colon Y\to G/B$ the corresponding closed
inclusion. The maps $i$ and $j$ give an open-closed decomposition of $G/B$.
The open-closed distinguished triangle

(4.74) $j_{!}j^{\ast}\widetilde{\mathcal{L}}\to\widetilde{\mathcal{L}}\to
i_{\ast}i^{\ast}\widetilde{\mathcal{L}}\overset{+1}{\longrightarrow}$

gives the distinguished triangle

(4.75)
$\pi_{s}^{\ast}\pi_{s\ast}j_{!}j^{\ast}\widetilde{\mathcal{L}}\to\pi_{s}^{\ast}\pi_{s\ast}\widetilde{\mathcal{L}}\to\pi_{s}^{\ast}\pi_{s\ast}i_{\ast}i^{\ast}\widetilde{\mathcal{L}}\overset{+1}{\longrightarrow}$

under an application of the triangulated functor $\pi_{s}^{\ast}\pi_{s\ast}$.
A further application of the triangulated functor
$k^{\prime}_{!}k^{\prime\ast}$ yields the distinguished triangle

(4.76)
$k^{\prime}_{!}k^{\prime\ast}\pi_{s}^{\ast}\pi_{s\ast}j_{!}j^{\ast}\widetilde{\mathcal{L}}\to
k_{!}^{\prime}k^{\prime\ast}\pi_{s}^{\ast}\pi_{s\ast}\widetilde{\mathcal{L}}\to
k_{!}^{\prime}k^{\prime\ast}\pi_{s}^{\ast}\pi_{s\ast}i_{\ast}i^{\ast}\widetilde{\mathcal{L}}\overset{+1}{\longrightarrow}.$

Simplifying, we obtain

(4.77)
$k_{!}^{\prime}(\pi_{s}k^{\prime})^{\ast}(\pi_{s}j)_{!}j^{\ast}\widetilde{\mathcal{L}}\to
k_{!}^{\prime}k^{\prime\ast}\pi_{s}^{\ast}\pi_{s\ast}\widetilde{\mathcal{L}}\to
k^{\prime}_{!}(\pi_{s}k^{\prime})^{\ast}(\pi_{s}i)_{\ast}i^{\ast}\widetilde{\mathcal{L}}\overset{+1}{\longrightarrow}.$

The diagram

${Ky_{0}B\times_{B}P_{s}/B}$${KyP_{s}/B}$${G/B}$${U}$${G/P_{s}}$${Ky_{0}B/B}$${G/B}$$\scriptstyle{\pi}$$\scriptstyle{\mu}$$\scriptstyle{k^{\prime}}$$\scriptstyle{\pi_{s}k^{\prime}}$$\scriptstyle{j}$$\scriptstyle{\pi_{s}j}$$\scriptstyle{j^{\prime}}$

shows

(4.78)
$k_{!}^{\prime}(\pi_{s}k^{\prime})^{\ast}(\pi_{s}j)_{!}j^{\ast}\widetilde{\mathcal{L}}=k^{\prime}_{!}\mu_{!}\pi^{\ast}j^{\prime\ast}\widetilde{\mathcal{L}},$

since the upper right square is cartesian. A similar diagram shows

(4.79)
$k_{!}^{\prime}(\pi_{s}k^{\prime})^{\ast}(\pi_{s}i)_{\ast}i^{\ast}\widetilde{\mathcal{L}}=k^{\prime}_{!}\mu_{!}\pi^{\ast}i^{\prime\ast}\widetilde{\mathcal{L}},$

where $i^{\prime}\,\colon\coprod_{i}Ky_{i}B/B\to G/B$. By Lemma 4.2.1, we have

(4.80)
$\begin{split}m(k_{!}^{\prime}\mu_{!}\pi^{\ast}j^{\prime\ast}\widetilde{\mathcal{L}})&=m(j^{\prime}_{!}j^{\prime\ast}\widetilde{\mathcal{L}})(T_{s}+1)\\\
m(k_{!}^{\prime}\mu_{!}\pi^{\ast}i^{\prime\ast}\widetilde{\mathcal{L}})&=m(i^{\prime}_{!}i^{\prime\ast}\widetilde{\mathcal{L}})(T_{s}+1)\end{split}$

since restricting to a $K$-orbit gives a direct sum of irreducible
$K$-equivariant local systems. By (4.10), (4.77), and (4.79), we have

(4.81)
$m(k^{\prime}_{!}k^{\prime\ast}\pi_{s}^{\ast}\pi_{s\ast}\widetilde{\mathcal{L}})=(m(j^{\prime}_{!}j^{\prime\ast}\widetilde{\mathcal{L}})+m(i^{\prime}_{!}i^{\prime\ast}\widetilde{\mathcal{L}}))(T_{s}+1)$

since $k_{!}^{\prime}\mu_{!}\pi^{\ast}i^{\prime\ast}\widetilde{\mathcal{L}}$
is $\ast$-even (q.v. Definition 7.2.8), as seen by observing that $\mu$ is
finite in (4.79) so $\mu_{!}$ is exact. It follows from (4.10) that

(4.82)
$(m(j^{\prime}_{!}j^{\prime\ast}\widetilde{\mathcal{L}})+m(i^{\prime}_{!}i^{\prime\ast}\widetilde{\mathcal{L}}))(T_{s}+1)=m(k^{\prime}_{!}k^{\prime\ast}\widetilde{\mathcal{L}})(T_{s}+1)$

by applying $k_{!}^{\prime}$ to the open-closed decomposition in $KyP_{s}/B$
of $k^{\prime\ast}\widetilde{\mathcal{L}}$. Therefore

(4.83)
$m(\pi_{s}^{\ast}\pi_{s\ast}\widetilde{\mathcal{L}})=\sum_{k^{\prime}}m(k^{\prime}_{!}k^{\prime\ast}\pi_{s}^{\ast}\pi_{s\ast}\widetilde{\mathcal{L}})=m(\widetilde{\mathcal{L}})(T_{s}+1)$

as desired.

Recall that our initial character map $\operatorname{ch}$ is given in (4.5).
We prove in Lemma 4.4.2 below that
$\pi_{s}^{\ast}\pi_{s\ast}\widetilde{\mathcal{L}}$ is $\ast$-even, so by (4.7)

(4.84)
$\begin{split}\operatorname{ch}(\pi_{s}^{\ast}\pi_{s\ast}\widetilde{\mathcal{L}})&=m(\pi_{s}^{\ast}\pi_{s\ast}\widetilde{\mathcal{L}})\\\
&=m(\widetilde{\mathcal{L}})(T_{s}+1)\\\
&=\operatorname{ch}(\widetilde{\mathcal{L}})(T_{s}+1)\end{split}$

since $\widetilde{\mathcal{L}}$ is $\ast$-even (q.v. Remark 4.4.4). The
theorem then follows from the fact that for any $\mathcal{F}\in D_{K}^{b}(X)$,

(4.85)
$\operatorname{ch}(\mathcal{F}[1])=q^{-1/2}\operatorname{ch}(\mathcal{F}).$

We note that $m$ is required to obtain additivity on certain distinguished
triangles (q.v. (4.10)) but $m$ fails to intertwine Hecke actions on odd
shifts of $\widetilde{\mathcal{L}}$, so we must move to $\operatorname{ch}$ to
obtain a character map which intertwines Hecke actions on the entire
semisimple category.

###  4.4. Application: Resolutions of singularities

We give a formula to compute the cohomology of fibers of the resolutions of
singularities defined in [Lar20].

For a subset $I\subseteq S$, we denote by $w_{I}$ the longest element of
$W_{I}$, $P_{I}=\overline{Bx_{I}B}$ the corresponding standard parabolic
subgroup, and $p_{I}$ the Poincaré polynomial of $P_{I}/B$ with indeterminate
$q^{1/2}$. Let $(W,\star)$ be the Richardson-Springer monoid associated to any
Coxeter group $(W,S)$, and recall the monoid action on $K\backslash G/B$ (cf.
[RS90]). For $x_{0}\in K\backslash G/B$, for $x_{1},\ldots,x_{\ell}\in
B\backslash G/B$, and for $1\leq i\leq\ell$, choose

(4.86) $J_{i}\subseteq\tau(x_{i-1})\cap\tau(x_{i}^{-1}),$

where

(4.87) $\tau(x_{i})=\left\\{s\in S\mid x_{i}\star s=x_{i}\right\\}.$

Given a subset $J\subseteq\tau(x_{\ell})$, choose $I\supseteq J$ and set
$x:=x_{0}\star x_{1}\star\cdots\star x_{\ell}\star x_{I}$. Set

(4.88)
$Z=\overline{Kx_{0}B}\times_{P_{J_{1}}}\overline{Bx_{1}B}\times_{P_{J_{2}}}\cdots\times_{P_{J_{\ell}}}\overline{Bx_{\ell}B}/P_{J}$

and define $\xi\,\colon Z\to G/P_{I}$ by

(4.89) $\xi[g_{0},\ldots,g_{\ell}]=g_{0}\cdots g_{\ell}P_{I}/P_{I}.$

Then $\xi$ is a $K$-equivariant proper map with image $\overline{KxB}/P_{I}$.

Recall that $m\,\colon D^{b}_{K}(G/B)\to M_{LV}$ is given by (4.6), and let
$h\,\colon D^{b}_{B}(G/B)\to\bm{H}$ be the corresponding character map for
$D^{b}_{G}(G\times_{G}G/B)=D^{b}_{B}(G/B)$ (as in [Soe00]).

######  Theorem 4.4.1.

Suppose $\overline{Kx_{0}B/B}$ is $\mathbb{Q}$-smooth, and for every $1\leq
i\leq\ell$, $\overline{Bx_{i}B/B}$ is $\mathbb{Q}$-smooth. Then for every
$y\in\overline{KxP_{I}/P_{I}}$, the Poincaré polynomial of $\xi^{-1}(y)$ with
indeterminate $q^{1/2}$ is computed by summing the coefficients of the local
systems on $KyB/B$ in the expression

(4.90)
$m(\widetilde{\mathbb{Q}}_{Kx_{0}B/B})\left(\prod_{i=1}^{\ell}h(\widetilde{\mathbb{Q}}_{Bx_{i}B/B})\
\right)h(\widetilde{\mathbb{Q}}_{P_{I}/B})\ p^{-1}$

in $M_{LV}$, where $p=p_{J}\displaystyle\prod_{i=1}^{\ell}p_{J_{i}}$.

The cohomology of the fiber $\xi^{-1}(y)$ is given by the stalk at $y$ of the
direct image $\xi_{\ast}\mathbb{Q}_{Z}$, and Theorem 4.4.1 states that these
stalks are determined by $m(\xi_{\ast}\mathbb{Q}_{Z})$. For every
$\mathcal{L}\in\mathscr{D}$ and $s\in S$ we prove in Lemma 4.4.2 that
$\pi_{s}^{\ast}\pi_{s\ast}\widetilde{\mathcal{L}}$ is $\ast$-even (q.v.
Definition 7.2.8). It follows that $\xi_{\ast}\mathbb{Q}_{Z}$ is $\ast$-even
since the pull-back of $\xi_{\ast}\mathbb{Q}_{Z}$ to $G/B$ is a direct summand
of even shifts of

(4.91)
$\pi_{s_{n}}^{\ast}\pi_{s_{n}\ast}\cdots\pi_{s_{1}}^{\ast}\pi_{s_{1}\ast}\widetilde{\mathcal{L}}$

for some choice of $\mathcal{L}$ and $s_{i}$. Stalks of any $\ast$-even sheaf
can be recovered from its image under $m$ by (4.9), so the cohomology of
$\xi^{-1}(y)$ is computed by $m(\xi_{\ast}\mathbb{Q}_{Z})$.

The explicit formula for $m(\xi_{\ast}\mathbb{Q}_{Z})$ in (4.90) comes from
first describing the direct image $\xi^{\prime\prime}_{\ast}\mathbb{Q}_{Z}$ in
terms of convolution of $\mathcal{H}$ on $\mathcal{M}_{LV}$, where

(4.92)
${\overline{Kx_{0}B}\times_{B}\overline{Bx_{1}B}\times_{B}\cdots\times_{B}\overline{Bx_{\ell}B}\times_{B}P_{I}/B}$${G/B}$${\overline{Kx_{0}B}\times_{P_{J_{1}}}\overline{Bx_{1}B}\times_{P_{J_{2}}}\cdots\times_{P_{J_{\ell}}}\overline{Bx_{\ell}B}\times_{P_{J}}P_{I}/B}$${G/B}$$\scriptstyle{\xi^{\prime\prime}}$$\scriptstyle{\zeta}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\xi^{\prime}}$

is such that $\xi^{\prime}$ is pullback of $\xi$ along $G/B\to G/P_{I}$. The
smooth map $\zeta$ has fiber $P_{J_{1}}/B\times\cdots\times
P_{J_{\ell}}/B\times P_{J}/B$, which explains why we divide by $p$ in (4.90)
(cf. [Wil11, Eq. (2.9)]). From this discussion, all that remains to prove
Theorem 4.4.1 is to establish that
$\pi_{s}^{\ast}\pi_{s\ast}\widetilde{\mathcal{L}}$ is $\ast$-even. We do this
in Lemma 4.4.2 below.

For every $(KxB/B,\mathcal{L})\in\mathscr{D}$, define

(4.93) $\tau(\mathcal{L})=\left\\{s\in S\mid x\star s=x,\ \mathcal{L}\text{
extends to }KxP_{s}/B\right\\}$

as in [Vog82, Def. 5.6], called the _Borho-Jantzen-Duflo $\tau$-invariant_.

######  Lemma 4.4.2.

For every $\mathcal{L}\in\mathscr{D}$ we have

(4.94) $\pi_{s}^{\ast}\pi_{s\ast}\widetilde{\mathcal{L}}\in
D^{b}_{K}(G/B)^{\ast-even}.$

###### Proof.

If $s\in\tau(\mathcal{L})$, then
$\mathcal{L}=\pi_{s}^{\ast}\mathcal{L}^{\prime}$ for some irreducible
$K$-equivariant local system $\mathcal{L}^{\prime}$ extended by zero to
$G/P_{s}$. By [ABV92, Prop. 7.15], it follows that
$\widetilde{\mathcal{L}}=\pi_{s}^{\ast}\widetilde{\mathcal{L}^{\prime}}$, so
(4.94) holds true by Lemma 4.2.4. If
$s\in\tau(x)\smallsetminus\tau(\mathcal{L})$, then
$\pi_{s\ast}\widetilde{\mathcal{L}}=0$ by restricting to $KxP_{s}/P_{s}$.
Otherwise $\pi_{s}$ restricts to a generically finite map on the support of
$\widetilde{\mathcal{L}}$.

Assume from now on that $\pi_{s}$ restricts to a generically finite map on the
support of $\widetilde{\mathcal{L}}$. We claim that
$\pi_{s\ast}\bm{IC}(\overline{\mathcal{O}},\mathcal{L})$ is perverse – even
though the direct image of $\bm{IC}(\overline{\mathcal{O}},\mathcal{L})$ under
a semi-small map need not be perverse. This will enable us to provide an upper
bound for possible terms occurring in the decomposition theorem, and moreover
will force each term to be shifted by zero.

Given $y\in\overline{KxB}$ let $k^{\prime}\,\colon KyP_{s}/P_{s}\to G/P_{s}$
be inclusion. The cartesian diagram

(4.95)
${KyP_{s}/B\cap\overline{KxB/B}}$${KyP_{s}/P_{s}}$${\overline{KxB/B}}$${G/P_{s}}$$\scriptstyle{\pi_{s}^{\prime}}$$\scriptstyle{k^{\prime\prime}}$$\scriptstyle{k^{\prime}}$$\scriptstyle{\pi_{s}\bar{k}}$

gives

(4.96)
$k^{\prime\ast}\pi_{s\ast}\bm{IC}(\overline{\mathcal{O}},\mathcal{L})=\pi_{s\ast}^{\prime}k^{\prime\prime\ast}\bm{IC}(\overline{\mathcal{O}},\mathcal{L})$

by base change. Equation (4.96) allows us to relate cohomology sheaves of the
direct image to the direct image over each stratum.

The following picture illustrates $\widetilde{\mathcal{L}}$ on the left and
$\pi_{s\ast}\widetilde{\mathcal{L}}$ on the right. The blue and red symbols
indicate possible nonzero cohomology sheaves when restricted to orbits. Note
that $\widetilde{\mathcal{L}}$ has zero odd cohomology sheaves by Remark
4.4.4.

The origin maps to the origin under $\pi_{s\ast}$ since we are assuming
$\pi_{s}^{\prime}$ (4.95) is finite over the open orbit; i.e., the left spade
is always mapped to the right spade. The other red symbols need not map as
depicted, but their locations on the right indicate an upper bound (with
respect to the diagonal) for nonzero cohomology sheaves. The red symbols on
the right represent all coordinates on the diagonal of
$\pi_{s\ast}\widetilde{\mathcal{L}}$ with possible nonzero cohomology sheaves.
This can occur when the coordinate for $\widetilde{\mathcal{L}}$ is distance
$\sqrt{2}$ from the diagonal and the map $\pi_{s}^{\prime}$ (4.95) has one
dimensional fiber (which contracts the stratum by one dimension). In
particular, every possible nonzero term on the diagonal has even degree. There
are no nonzero terms above the diagonal, so it follows that
$\pi_{s\ast}\bm{IC}(\overline{\mathcal{O}},\mathcal{L})$ is perverse such that

(4.97)
$\pi_{s\ast}\bm{IC}(\overline{\mathcal{O}},\mathcal{L})=\bigoplus_{\mathcal{L}^{\prime}}\bm{IC}(\overline{\mathcal{O}^{\prime}},\mathcal{L}^{\prime}),$

where the dimension of all
$\mathcal{O}^{\prime}\subseteq\overline{KxP_{s}/P_{s}}$ appearing in the sum
have same parity. The claim follows by shifting degree of (4.97) by
$-\dim\mathcal{O}$. ∎

######  Definition 4.4.3.

An element $(\mathcal{O},\mathcal{L})\in\mathscr{D}$ is _clean_ if

(4.98) $\widetilde{\mathcal{L}}=k_{\mathcal{O}!}\mathcal{L}.$

######  Remark 4.4.4.

Every block (q.v. Definition 5.1.1) is generated by clean local systems, which
are $\ast$-even since their cohomology sheaves vanish outside of degree zero.
It follows that the proof of Lemma 4.4.2 can also be used to show the classic
fact [LV83, Thm. 1.12(a)] that $\widetilde{\mathcal{L}}\in
D_{K}^{b}(G/B)^{\ast-even}$.

##  5\. A categorification of the trivial block

In Section 4, we provide a geometric categorification of the entire
Lusztig–Vogan module $M_{LV}$. By considering the subcategory of
$\mathcal{M}_{LV}^{ss}$ generated by trivial $K$-equivariant local systems on
closed orbits, we can refine this categorification to give a categorification
of the trivial block of $M_{LV}$. The purpose of considering this refinement
is that on the subcategory generated by closed orbits, the hypercohomology
functor is fully faithful (Theorem 7.2.1), which allows us to use Soergel-type
techniques to describe the category algebraically. In this section, we recall
the notion of blocks and give a geometric categorification of the trivial
block.

###  5.1. Blocks

There is a natural equivalence relation on the set $\mathscr{D}$ in (3.1)
which partitions $M_{LV}$ into blocks. In this section we introduce a
geometric formulation of this equivalence relation which is equivalent to the
original algebraic formulation in [Vog81, Def. 9.2.1], see [Vog83].

Let $G,B,K$, and $X$ be as in Section 1.2. For a parameter
$(\mathcal{O},\mathcal{L})\in\mathscr{D}$, the cohomology sheaves
$\mathcal{H}^{i}(\bm{IC}(\overline{\mathcal{O}},\mathcal{L}))$ are objects in
the abelian category of $K$-equivariant constructible sheaves on $X$. We
denote by
$[\mathcal{L}^{\prime}:\mathcal{H}^{i}(\bm{IC}(\overline{\mathcal{O}},\mathcal{L})]$
the multiplicity of
$\mathcal{L}^{\prime}\in\operatorname{Loc}_{K}(\mathcal{O}^{\prime})$
(considered as a $K$-equivariant constructible sheaf on $X$ by extending by
zero) in the composition series of
$\mathcal{H}^{i}(\bm{IC}(\overline{\mathcal{O}},\mathcal{L}))$.

######  Definition 5.1.1.

Block equivalence is the equivalence relation on $\mathscr{D}$ generated by

$(\mathcal{O},\mathcal{L})\sim(\mathcal{O}^{\prime},\mathcal{L}^{\prime})\text{
if
}[\mathcal{L}^{\prime}:\mathcal{H}^{i}(\bm{IC}(\overline{\mathcal{O}},\mathcal{L})]\neq
0$

for some $i\in\mathbb{Z}$. Equivalence classes under $\sim$ are called blocks.

######  Remark 5.1.2.

We have formulated block equivalence to be the equivalence relation generated
by non-zero KLV polynomials [LV83, Thm. 1.12]. There are also equivalent
formulations using Ext groups of $(\mathfrak{g},K)$-modules [Vog81, Def.
9.2.1] and composition series of standard modules [Vog83, Def. 1.14].

The trivial local system on the open $K$-orbit is the Lusztig–Vogan parameter
corresponding to the trivial representation of $G_{\mathbb{R}}$. Because of
this, we refer to this block containing the trivial local system on the open
orbit as the block of the trivial representation, and denote it
$\mathscr{D}^{0}$. We focus our attention on this block for several reasons.
One basic reason is that it is natural to study representations with the
trivial representation as a composition factor. Another reason is that
$\mathcal{D}^{0}$ is the block that governs birational resolutions of
$K$-orbit closures, see Section 4.4.

The following standard (but difficult) fact is vital to our arguments. This
follows from [Vog81, Thm. 9.2.11] and the algorithm for computing KLV
polynomials in [Vog82, LV83]. A geometric proof can also be found in [BV21,
Prop. 3.6], and another proof using Vogan duality is given in [BV21, Rmk.
3.7].

######  Theorem 5.1.3.

Let $M_{LV}^{0}$ be the $\bm{H}$-module generated by the action of the
operators $T_{s},s\in S$ (Definition 3.2.1) on
$\\{(\mathcal{Q},\mathbb{Q}_{\mathcal{Q}})\mid\mathcal{Q}\text{ closed
$K$-orbit}\\}\subseteq\mathscr{D}$. Then

$M_{LV}^{0}=\bigoplus_{\mathcal{L}\in\mathscr{D}^{0}}\mathbb{Z}[q^{\pm
1}]\mathcal{L}.$

######  Example 5.1.4.

$($Blocks for $SL_{2}(\mathbb{R}))$ We continue with the setup in Example
3.2.4. The $\bm{IC}$ sheaf corresponding to the parameter
$(\mathcal{O},\mathbb{Q}_{\mathcal{O}})\in\mathscr{D}$ is
$\bm{IC}(X,\mathbb{Q}_{\mathcal{O}})=\mathbb{Q}_{X}[1]$, which has cohomology
sheaves

$\mathcal{H}^{i}(\bm{IC}(X,\mathbb{Q}_{\mathcal{O}}))=\begin{cases}\mathbb{Q}_{X}&i=-1;\\\
0&\text{ else}.\end{cases}$

Both $\mathbb{Q}_{\mathcal{Q}_{0}}$ and $\mathbb{Q}_{\mathcal{Q}_{\infty}}$
(extended by $0$ and considered as $K$-equivariant constructible sheaves on
$X$) are subsheaves of $\mathbb{Q}_{X}$, so

$(\mathcal{Q}_{0},\mathbb{Q}_{\mathcal{Q}_{0}})\sim(\mathcal{Q}_{\infty},\mathbb{Q}_{\mathcal{Q}_{\infty}})\sim(\mathcal{O},\mathbb{Q}_{\mathcal{O}})$

in the equivalence relation of Definition 5.1.1. As the Möbius band local
system $\mathcal{L}$ does not extend (as a local system) to $X$, the parameter
$(\mathcal{O},\mathcal{L})$ is not in the same block as any other
Lusztig–Vogan parameter. Hence $\mathscr{D}$ has two blocks,

$\mathscr{D}^{0}=\\{(\mathcal{Q}_{0},\mathbb{Q}_{\mathcal{Q}_{0}}),(\mathcal{Q}_{\infty},\mathbb{Q}_{\mathcal{Q}_{\infty}}),(\mathcal{O},\mathbb{Q}_{\mathcal{O}})\\},\text{
and }\mathscr{D}^{1}=\\{(\mathcal{O},\mathcal{L})\\}.$

We can see from the formulas (3.7) that the submodule of $M_{LV}$ generated by
trivial local systems on closed orbits is spanned as a $\mathbb{Z}[q^{\pm
1}]$-module by $\mathscr{D}^{0}$.

###  5.2. The category generated by closed orbits

Let $\mathcal{Q}$ be a closed $K$-orbit in $X$ and
$i_{\mathcal{Q}}:\mathcal{Q}\rightarrow X$ the inclusion map. Denote by

(5.1)
$\bm{IC}_{\mathcal{Q}}:=\bm{IC}(\mathcal{Q},\mathbb{Q}_{\mathcal{Q}})=i_{\mathcal{Q}!}\mathbb{Q}_{\mathcal{Q}}[\dim\mathcal{Q}]=i_{\mathcal{Q}*}\mathbb{Q}_{\mathcal{Q}}[\dim\mathcal{Q}]$

the $\bm{IC}$ sheaf corresponding to the trivial $K$-equivariant local system
on $\mathcal{Q}$.

######  Definition 5.2.1.

Let $\mathcal{M}_{LV}^{0}$ be the full subcategory of $D^{b}_{K}(X)$ generated
by $\\{\bm{IC}_{\mathcal{Q}}\mid\mathcal{Q}\text{ closed}\\}$ under right
convolution by objects in $\mathcal{H}$, finite direct sums ($\oplus$), direct
summands ($\ominus$), and grading shifts ($[1]$). That is,

$\mathcal{M}_{LV}^{0}:=\left\langle\bm{IC}_{\mathcal{Q}}*\mathcal{H}\mid\mathcal{Q}\text{
closed}\right\rangle_{\oplus,\ominus,[1]}.$

By construction, the category $\mathcal{M}^{0}_{LV}$ is a right
$\mathcal{H}$-module, hence the split Grothendieck group
$[\mathcal{M}_{LV}^{0}]_{\oplus}$ has the structure of a right
$\bm{H}$-module.

######  Theorem 5.2.2.

Let $\mathcal{M}_{LV}^{0}$ be as in Definition 5.2.1 and $M_{LV}^{0}$ as in
Theorem 5.1.3. As right $\bm{H}$-modules,

$[\mathcal{M}_{LV}^{0}]_{\oplus}\simeq M_{LV}^{0}$

###### Proof.

This follows immediately from Theorem 5.1.3 and Theorem 4.1.3. ∎

######  Example 5.2.3.

$($The category $\mathcal{M}_{\mathrm{LV}}^{0}$ for $SL_{2}(\mathbb{R}))$ We
return to the setup of Examples 3.2.4 and 4.2.9. To compute the action of
$\mathcal{H}$ we use Lemma 4.1.2. Because $\mathcal{Q}_{0}$ and
$\mathcal{Q}_{\infty}$ are closed, we have
$\bm{IC}(X,\mathbb{Q}_{\mathcal{Q}_{0}})=i_{0*}\mathbb{Q}_{\mathcal{Q}_{0}}$
and
$\bm{IC}(X,\mathbb{Q}_{\mathcal{Q}_{\infty}})=i_{\infty*}\mathbb{Q}_{\mathcal{Q}_{\infty}}$.
So by (4.56), we have

(5.2) $\displaystyle\bm{IC}_{\mathcal{Q}_{0}}*\bm{IC}_{s}$
$\displaystyle=\pi_{s}^{*}\pi_{s*}\bm{IC}_{\mathcal{Q}_{0}}[1]$
$\displaystyle=\mathbb{Q}_{X}[1]$
$\displaystyle=\bm{IC}(X,\mathbb{Q}_{\mathcal{O}}).$

Similarly,

(5.3)
$\bm{IC}_{\mathcal{Q}_{\infty}}*\bm{IC}_{s}=\bm{IC}(X,\mathbb{Q}_{\mathcal{O}}).$

Using Lemma 4.2.4, we compute

(5.4) $\displaystyle\bm{IC}(X,\mathbb{Q}_{\mathcal{O}})*\bm{IC}_{s}$
$\displaystyle=\pi_{s}^{*}\pi_{s*}\mathbb{Q}_{X}[2]$
$\displaystyle=\pi_{s}^{*}(\mathbb{Q}_{\mathrm{pt}}\oplus\mathbb{Q}_{\mathrm{pt}}[2])$
$\displaystyle=\mathbb{Q}_{X}\oplus\mathbb{Q}_{X}[2]$
$\displaystyle=\bm{IC}(X,\mathbb{Q}_{\mathcal{O}})[-1]\oplus\bm{IC}(X,\mathbb{Q}_{\mathcal{O}})[1].$

We conclude from these computations that

$\mathcal{M}_{\mathrm{LV}}^{0}=\langle\bm{IC}_{\mathcal{Q}_{0}},\bm{IC}_{\mathcal{Q}_{\infty}},\bm{IC}(X,\mathbb{Q}_{\mathcal{O}})\rangle_{\oplus,[1]}.$

######  Example 5.2.4.

$($The non-trivial block for $SL_{2}(\mathbb{R}))$ In the setting of Example
5.2.3, it is also interesting to compute the action of $\mathcal{H}$ on the
$\bm{IC}$ sheaf corresponding to the Möbius band local system $\mathcal{L}$.
First, note that $\mathcal{L}$ is clean (q.v. Definition 4.4.3), so

$\bm{IC}(X,\mathcal{L})=j_{!*}\mathcal{L}=j_{!}\mathcal{L}=j_{*}\mathcal{L},$

where $j:\mathcal{O}\hookrightarrow X$ is inclusion. Hence by the computation
of $H^{*}(\mathbb{C}^{\times};\mathcal{L})$ in Example 4.2.10, we have

(5.5)
$\bm{IC}(X,\mathcal{L})*\bm{IC}_{s}=\pi_{s}^{*}\pi_{s*}j_{!}\mathcal{L}[2]=0.$

##  6\. An algebraic categorification

In this section, we introduce a category of bimodules which plays the role of
Soergel bimodules in the $K$-equivariant setting. Before introducing our
category, we recall some facts about torus fixed points in $K$-orbits which
will be essential in the arguments of Section 7.

###  6.1. Torus fixed points and closed $K$-orbits

For the remainder of the paper, we impose an additional connectedness
assumption on the group $K$. Let $G$ be as in Section 1, and set $K$ to be the
identity component of the fixed point group $G^{\theta}$. With this
assumption, the group $K$ has the structure of a connected reductive complex
algebraic group. Denote by $\mathfrak{g}$ and $\mathfrak{k}$ the corresponding
Lie algebras. The involution $\theta$ induces an involution on $\mathfrak{g}$
which we refer to by the same symbol. Let

$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$

be the decomposition of $\mathfrak{g}$ into eigenspaces of $\theta$. Here
$\mathfrak{k}$ is the $+1$-eigenspace and $\mathfrak{p}$ is the
$-1$-eigenspace.

######  Remark 6.1.1.

The assumption that $K$ is connected is necessary for several of the arguments
in Section 7. We suspect that the main results of this paper remain true for
disconnected $K$, but to prove them one would need to alter the present
arguments. We intend to explore this further in future work.

Choose a maximal torus $T_{K}$ in $K$, and let $W_{K}:=N_{K}(T_{K})/T_{K}$ be
the Weyl group of $(K,T_{K})$. Set

(6.1) $T:=Z_{G}(T_{K}).$

######  Lemma 6.1.2.

The group $T$ is the unique $\theta$-stable maximal torus containing $T_{K}$.

###### Proof.

Clearly $T$ is $\theta$-stable: for $z\in T$ and $t\in T_{K}$,

$\theta(z)t=\theta(zt)=\theta(tz)=t\theta(z),$

so $\theta(z)\in T$.

As $T_{K}$ is a $\theta$-fixed maximal torus in $K$, it is contained in some
$\theta$-stable maximal torus $T^{\prime}$ of $G$. Let
$\Sigma=\Sigma(\mathfrak{g},\mathfrak{t}^{\prime})$ be the root system of
$(\mathfrak{g},\mathfrak{t}^{\prime}:=\operatorname{Lie}T^{\prime})$, and

$\mathfrak{g}=\mathfrak{t}^{\prime}\oplus\bigoplus_{\alpha\in\Sigma}\mathfrak{g}_{\alpha}$

the corresponding root space decomposition of $\mathfrak{g}$. The Lie algebra
$\mathfrak{t}^{\prime}$ decomposes into a direct sum
$\mathfrak{t}^{\prime}=\mathfrak{t}_{K}\oplus\mathfrak{a}$, where
$\mathfrak{t}_{K}=\operatorname{Lie}T_{K}$ and
$\mathfrak{a}=\mathfrak{t}^{\prime}\cap\mathfrak{p}$. Recall that a root
$\alpha\in\Sigma$ is said to be real if $\alpha|_{\mathfrak{t}_{K}}=0$. If
$X\in\mathfrak{g}_{\alpha}$ for $\alpha$ real, then $[t,x]=\alpha(t)x=0$ for
all $t\in\mathfrak{t}_{K}$. Hence we have a decomposition

(6.2)
$Z_{\mathfrak{g}}(\mathfrak{t}_{K})=\mathfrak{t}^{\prime}\oplus\bigoplus_{\alpha\in\Sigma\atop{\alpha\text{
real}}}\mathfrak{g}_{\alpha}.$

Because $T_{K}$ is a maximal torus in $K$, the Cartan subalgebra
$\mathfrak{t}^{\prime}$ is maximally compact, meaning that the compact
dimension (i.e., the dimension of the $+1$ $\theta$-eigenspace) is maximal
among $\theta$-stable Cartan subalgebras. The maximally compact Cartan
subalgebras of $\mathfrak{g}$ are precisely those with no real roots [Kna94,
Prop. 6.70], so in our case, (6.2) becomes

$Z_{\mathfrak{g}}(\mathfrak{t}_{K})=\mathfrak{t}^{\prime}.$

This implies the group-theoretic statement. ∎

Let $W:=N_{G}(T)/T$ be the Weyl group of $(G,T)$. The following lemma shows
that $W$ contains $W_{K}$ as a subgroup.

######  Lemma 6.1.3.

There is an embedding $W_{K}\hookrightarrow W$.

###### Proof.

To begin, we claim that

(6.3) $N_{K}(T)=N_{K}(T_{K}).$

Indeed, let $k\in N_{K}(T)$ and $t\in T_{K}$. Then there exists $t^{\prime}\in
T$ such that $kt=t^{\prime}k$. As both elements $kt$ and $k$ are fixed by
$\theta$, we have

$t^{\prime}k=\theta(t^{\prime}k)=\theta(t^{\prime})k,$

so $t^{\prime}$ is also fixed by $\theta$. Hence $N_{K}(T)\subseteq
N_{K}(T_{K})$.

Conversely, consider $kTk^{-1}$ for $k\in N_{K}(T_{K})$. This is a
$\theta$-stable maximal torus in $G$ containing $T_{K}$, so by Lemma 6.1.2,
$kTk^{-1}=T$. Hence $N_{K}(T_{K})\subseteq N_{K}(T)$.

Hence there is a natural injection

$N_{K}(T_{K})/T_{K}\xhookrightarrow{i}N_{G}(T)/T_{K}.$

By composing $i$ with the quotient map

$N_{G}(T)/T_{K}\xrightarrow{q}N_{G}(T)/T,$

we obtain a map

(6.4) $W_{K}=N_{K}(T_{K})/T_{K}\xrightarrow{i\circ q}N_{G}(T)/T=W.$

As $\ker q=T/T_{K}$ is not contained in the $\operatorname{im}i$, the map
(6.4) is injective. ∎

Because the torus $T$ is $\theta$-stable, $\theta$ induces an involution on
$W$. Let $W^{\theta}$ be the group of elements fixed by this involution. After
identifying $W_{K}$ with its image under the injection in Lemma 6.1.3, we have

(6.5) $W_{K}\subseteq W^{\theta}\subseteq W.$

Choose a Borel subgroup $B\subseteq G$ containing $T$, and set $B_{K}=B\cap
K$. The group $B_{K}$ is a Borel subgroup in $K$ containing $T_{K}$. Note that
$B$ is $\theta$-stable by construction. Let $S\subseteq W$ be the set of
simple reflections determined by the choice of $B$ and
$\Sigma^{+}\subseteq\Sigma$ be the corresponding set of positive roots.

It is well-known that the $T$-fixed points in the flag variety $X=G/B$ are in
bijection with $W$, and each Bruhat cell contains exactly one $T$-fixed point.
Specifically, we have

(6.6) $X^{T}=\\{x_{w}:=wB/B\\}_{w\in W},\text{ and }x_{w}\in X_{w}=B\cdot
wB/B.$

######  Lemma 6.1.4.

All $T_{K}$-fixed points in $X$ are fixed by $T$. In particular,
$X^{T_{K}}=X^{T}$.

###### Proof.

Each $T$-fixed point $x_{w}$ lies in an affine chart
$U_{w}\simeq\mathbb{A}^{|\Sigma^{+}|}$ on which $T$ acts linearly by roots. In
particular, we can identify

$x_{w}=0\in U_{w}=\bigoplus_{\alpha\in-w(\Sigma^{+})}\mathbb{C}_{\alpha},$

where $T$ acts on $\mathbb{C}_{\alpha}$ by $t\cdot z=\exp\alpha(t)z$ for $t\in
T$, $z\in\mathbb{C}_{\alpha}$. The charts $\\{U_{w}\\}_{w\in W}$ cover $X$.
Clearly $x_{w}$ is the only $T$-fixed point in $U_{w}$.

By restriction we obtain a linear action of $T_{K}$ on $U_{w}$, with weights

$\\{\widetilde{\alpha}:=\alpha|_{\mathfrak{t}_{K}}\\}_{\alpha\in-w(\Sigma^{+})}.$

The point $x_{w}\in U_{w}$ is fixed by $T_{K}$. Moreover, $x_{w}$ is the only
$T_{K}$-fixed point in $U_{w}$. To see this, recall from the proof of Lemma
6.1.2 that $\mathfrak{t}=\operatorname{Lie}T$ is maximally compact, so there
are no real roots in $\Sigma$ [Kna94, Prop. 6.70]. Hence the restriction map
$\mathfrak{t}^{*}\rightarrow\mathfrak{t}_{K}^{*}$ is non-zero on all roots
$\alpha\in\Sigma$, and thus the weights of the $T_{K}$ action on $U_{w}$ are
non-zero for all $w\in W$.

As the charts $U_{w}$ cover $X$ and each chart contains a single $T_{K}$-fixed
point which is also fixed by $T$, the lemma follows. ∎

######  Lemma 6.1.5.

Each closed $K$-orbit $\mathcal{Q}$ contains $|W_{K}|$ $T$-fixed points.
Moreover, if $x_{w}\in\mathcal{Q}$, then $x_{vw}\in\mathcal{Q}$ for all $v\in
W_{K}$, and all representatives of the coset $W_{K}w\in W_{K}\backslash W$ are
in $W^{\theta}$.

###### Proof.

Fix a closed $K$-orbit $\mathcal{Q}$. For any $q\in\mathcal{Q}$, the
stabilizer $K_{q}:=\operatorname{stab}_{K}(q)$ is solvable (as it is the
intersection of the solvable group $G_{q}$ with $K$), parabolic (as
$\mathcal{Q}\simeq K/K_{q}$ is a closed subvariety of a projective variety),
and connected (as it is a parabolic subgroup of a connected group), hence it
is a Borel subgroup of $K$. This implies that $\mathcal{Q}$ is isomorphic to
the flag variety of $K$, so it contains $|W_{K}|$ $T_{K}$-fixed points. By
Lemma 6.1.4, all of these must also be fixed by $T$.

There is a natural action of $W$ on $X^{T}$ coming from the conjugation action
of $N_{G}(T)$. For $v\in W$, this action sends $x_{w}\mapsto x_{vw}$.
Similarly, as $\mathcal{Q}$ is isomorphic to the flag variety of $K$, there is
an action of $W_{K}$ on $\mathcal{Q}^{T_{K}}=\mathcal{Q}^{T}$ coming from the
conjugation action of $N_{K}(T_{K})$. Under the identification of $W_{K}$ with
a subgroup of $W$ (Lemma 6.1.3), the action of $W_{K}$ on $\mathcal{Q}^{T}$ is
just the restriction of the $W$-action to $W_{K}$. This proves the first half
of the second statement.

To see that the $T$-fixed points in closed orbits correspond to elements in
$W^{\theta}$, we shift perspectives and identify $X$ with the variety of Borel
subalgebras of $\mathfrak{g}$. Under this identification, the $T$-fixed point
$x_{w}$ corresponds to the Borel subalgebra

$\mathfrak{b}^{w}:=\mathfrak{t}\oplus\bigoplus_{\alpha\in\Sigma^{+}}\mathfrak{g}_{w\alpha}.$

The involution $\theta:G\rightarrow G$ induces involutions on $\mathfrak{g}$,
$W$, and $\Sigma$. In what follows, we use the same symbol $\theta$ for each
of these involutions.

The closed $K$-orbits are precisely those consisting of $\theta$-stable Borel
subalgebras [Mil93, Lem. 5.8], so the Weyl group elements $w\in W$
corresponding to fixed points in closed orbits are exactly those $w$ such that
$\mathfrak{b}^{w}$ is $\theta$-stable. We have

$\displaystyle\theta(\mathfrak{b}^{w})$
$\displaystyle=\theta(\mathfrak{t})\oplus\bigoplus_{\alpha\in\Sigma^{+}}\theta(\mathfrak{g}_{w\alpha})$
$\displaystyle=\mathfrak{t}\oplus\bigoplus_{\alpha\in\Sigma^{+}}\mathfrak{g}_{\theta(w)\theta(\alpha)}$
$\displaystyle=\mathfrak{t}\oplus\bigoplus_{\alpha\in\Sigma^{+}}\mathfrak{g}_{\theta(w)\alpha}.$

Here the third equality follows from the fact that
$\theta(\Sigma^{+})=\Sigma^{+}$ because $\Sigma$ has no real roots [Kna94, Ch.
VI §8]. From this we see that $\theta(\mathfrak{b}^{w})=\mathfrak{b}^{w}$ if
and only if $\theta(w)=w$, so all $T$-fixed points in closed orbits correspond
to $w\in W^{\theta}$. This completes the proof of the second statement. ∎

Lemma 6.1.5 gives a useful combinatorial description of the closed $K$-orbits
in $X$.

######  Corollary 6.1.6.

The closed $K$-orbits are in bijection with $W_{K}\backslash W^{\theta}$.

######  Example 6.1.7.

$($Torus fixed points for $SL_{2}(\mathbb{R}))$ We return to our running
example (Examples 3.2.4, 4.2.9, 4.2.10, 5.1.4, 5.2.3, 5.2.4). In this case,
$T_{K}=T=K$, so $T$-fixed points are the closed orbits $\mathcal{Q}_{0}$ and
$\mathcal{Q}_{\infty}$. We have $W_{K}=\\{1\\}\subseteq W=S_{2}=W^{\theta}$.
Clearly $W_{K}\backslash W^{\theta}$ is in bijection with the set of closed
orbits.

###  6.2. A category of bimodules

We now describe the category which provides an algebraic categorification of
the trivial block of the Lusztig–Vogan module. We begin by introducing two
rings.

As above, denote by $X(T)_{\mathbb{Q}}:=X(T)\otimes_{\mathbb{Z}}\mathbb{Q}$
and $X(T_{K})_{\mathbb{Q}}:=X(T_{K})\otimes_{\mathbb{Z}}\mathbb{Q}$ the
$\mathbb{Q}$-span of the character lattices of $T$ and $T_{K}$. Let
$P:=S(X(T_{K})_{\mathbb{Q}})$ be the symmetric algebra of
$X(T_{K})_{\mathbb{Q}}$, graded so that $X(T_{K})_{\mathbb{Q}}$ has degree
$2$, and denote by

(6.7) $P^{K}:=P^{W_{K}}=\\{p\in P\mid wp=p\text{ for all }w\in W_{K}\\}.$

As in Section 3.3, set $R=S(X(T)_{\mathbb{Q}})$. Let

(6.8) $\phi:R\rightarrow P$

be the natural algebra homomorphism extending the restriction map

$X(T)\rightarrow X(T_{K}).$

Let $(P^{K},R)\mathrm{-gbim}$ be the category of finitely generated (as both
left $P^{K}$\- and right $R$-modules) graded $(P^{K},R)$-bimodules, with
morphisms given by graded $(P^{K},R)$-bimodule maps which are homogeneous of
degree $0$. Denote by
$(n):(P^{K},R)\mathrm{-gbim}\rightarrow(P^{K},R)\mathrm{-gbim}$ the natural
shift functor, as defined in Section 3.3. There is a right action of
$\mathbb{S}\text{Bim}$ on $(P^{K},R)\mathrm{-gbim}$ given by the tensor
product: for $M\in(P^{K},R)\mathrm{-gbim}$ and $S\in\mathbb{S}\text{Bim}$,

(6.9) $M\otimes_{R}S\in(P^{K},R)\mathrm{-gbim}$

To define our category, we imitate the geometric construction in Section 5.2.
First we need a class of generating objects to play the role of trivial local
systems on closed orbits.

######  Definition 6.2.1.

For every $x\in W$, define the corresponding standard bimodule
$P_{x}\in(P^{K},R)$-gbim as follows. As a vector space, $P_{x}=P$. The left
action of $f\in P^{K}$ is given by left multiplication in $P$:

$f\cdot p:=fp\text{ for $p\in P_{x}$}.$

The right action of $g\in R$ is given by right multiplication by $\phi(xg)$,
where $\phi$ is the algebra homomorphism (6.8):

$p\cdot_{x}g:=p\phi(xg)\text{ for $p\in P_{x}$}.$

######  Notation 6.2.2.

We write the action of $W$ on $R$ (or $W_{K}$ on $P$) with no symbol. It is
implied from this notation that an element $x\in W$ acts only on the element
immediately following it. If we wish for $x\in W$ to act on a product of
elements, we use parentheses. For example, $wrs$ denotes the product in $R$ of
the elements $wr$ and $s$, whereas $w(rs)$ denotes the image of the element
$rs$ under action by $w$. Note that because the action of $W$ on
$R=S(\mathfrak{t}^{*})$ (and that of $W_{K}$ on $P=S(\mathfrak{t}^{*}_{K})$)
is obtained by extending the natural reflection action of $W$ on
$\mathfrak{t}^{*}$ (resp. $W_{K}$ on $\mathfrak{t}_{K}^{*}$) multiplicatively,
we have $w(rs)=wrws$.

######  Lemma 6.2.3.

If $w\in W_{K}$,

$P_{wx}\simeq P_{x}$

as $(P^{K},R)$-bimodules.

###### Proof.

Let $\varphi:P_{wx}\rightarrow P_{x}$ be the linear map sending $p\mapsto
w^{-1}p$. We claim that $\varphi$ is an isomorphism of $(P^{K},R)$-bimodules.

Clearly $\varphi$ is a bijection of vector spaces. Let $f\in P^{K}$, $p\in
P_{wx}$. The following computation shows that $\varphi$ is a left
$P^{K}$-module morphism:

$f\cdot\varphi(p)=f\cdot
w^{-1}p=fw^{-1}p=w^{-1}fw^{-1}p=w^{-1}(fp)=\varphi(fp)=\varphi(f\cdot p).$

Let $g\in R$, $p\in P_{wx}$. The following computation shows that $\varphi$ is
a morphism of right $R$-modules:

$\displaystyle\varphi(p)\cdot_{x}g=w^{-1}p\cdot_{x}g=w^{-1}p\phi(xg)$
$\displaystyle=w^{-1}(pw\phi(xg))$
$\displaystyle=\varphi(pw\phi(xg))=\varphi(p\phi(wxg))=\varphi(p\cdot_{wx}g).$

This completes the proof. ∎

Recall from Section 6.1 that $W_{K}$ is contained in $W^{\theta}$, the
subgroup of $\theta$-fixed elements of $W$ (6.5). Choose a set
${}^{K}W^{\theta}$ of coset representatives for $W_{K}\backslash W^{\theta}$.
We have a set of distinguished objects

$\\{P_{x}\mid x\in{{}^{K}W^{\theta}}\\}\subseteq(P^{K},R)\mathrm{-gbim}$

parameterized by cosets $W_{K}\backslash W^{\theta}$. By Lemma 6.2.3 and Lemma
6.1.5, for every $w\in W$ such that the corresponding $T$-fixed point $x_{w}$
is in a closed $K$-orbit, $P_{w}$ is isomorphic to an element of this set.

######  Remark 6.2.4.

If $G_{\mathbb{R}}$ is equal rank, then the subgroup $W_{K}$ is a reflection
subgroup of $W$ and $W^{\theta}=W$. In this case, every right coset
$\overline{x}\in W_{K}\backslash W$ has a unique minimal-length representative
[Dye90, Cor. 3.4]. In contrast, for general $K$, it is not the case that each
right $W_{K}$-coset in $W^{\theta}$ has a minimal length representative.

######  Definition 6.2.5.

Let $\mathcal{N}_{LV}^{0}$ be the full subcategory of $(P^{K},R)$-gbim
generated by $\\{P_{x}\mid x\in{{}^{K}W^{\theta}}\\}$ under action (6.9) by
objects in $\mathbb{S}\text{Bim}$, finite direct sums ($\oplus$), direct
summands ($\ominus$), and grading shifts ($(1)$). That is,

$\mathcal{N}_{LV}^{0}:=\left\langle P_{x}\otimes_{R}\mathbb{S}\text{Bim}\mid
x\in{{}^{K}W^{\theta}}\right\rangle_{\oplus,\ominus,(1)}.$

By construction, $\mathcal{N}_{LV}^{0}$ is a right module over the monoidal
category $\mathbb{S}\text{Bim}$, in the sense of [EGNO16, Ch. 7]. Let
$[\mathcal{N}_{LV}^{0}]_{\oplus}$ be the split Grothendieck group of
$\mathcal{N}_{LV}^{0}$. Then by Theorem 3.3.4,
$[\mathcal{N}_{LV}^{0}]_{\oplus}$ has the structure of a right
$\bm{H}$-module.

######  Example 6.2.6.

$($The category $\mathcal{N}_{\mathrm{LV}}^{0}$ for $SL_{2}(\mathbb{R}))$ We
return to our running example (Examples 3.2.4, 4.2.9, 4.2.10, 5.1.4, 5.2.3,
5.2.4, 6.1.7). As $T_{K}=T$ and $W_{K}$ is trivial,

$P^{K}=P=R=\mathbb{Q}[\alpha]$