# Affine Harish-Chandra bimodules and Steinberg–Whittaker localization

Justin Campbell  California Institute of Technology
Pasadena, CA 91125, USA<EMAIL_ADDRESS>and  Gurbir Dhillon  Yale
University
New Haven, CT 06511, USA<EMAIL_ADDRESS>
###### Abstract.

We construct categories of Harish-Chandra bimodules for affine Lie algebras
analogous to Harish-Chandra bimodules with infinitesimal characters for simple
Lie algebras, addressing an old problem raised by I. Frenkel and Malikov.
Under an integrality hypothesis, we establish monoidal equivalences between
blocks of our affine Harish-Chandra bimodules and versions of the affine Hecke
category. To do so, we introduce a new singular localization onto Whittaker
flag manifolds. The argument for the latter specializes to a somewhat
alternative treatment of regular localization, and identifies the category of
Lie algebra representations with a generalized infinitesimal character as the
parabolic induction, in the sense of categorical representation theory, of a
Steinberg module.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and notation
  3. 3 Hecke algebras and invariant vectors
  4. 4 A category of Lie algebra representations
  5. 5 Localization I: The sign representation and the Steinberg module
  6. 6 parabolic induction for groups and Hecke categories
  7. 7 Localization II: the parabolic induction of the Steinberg module and Lie algebra representations
  8. 8 The linkage principle for positive energy representations
  9. 9 Some applications
  10. 10 Affine Harish-Chandra bimodules

##  1\. Introduction

### 1.1.

The study of Harish-Chandra bimodules with generalized infinitesimal
characters has been of central importance in the representation theory of
semisimple Lie algebras. As an early triumph, it identified categories of
principal series representations of complex Lie groups with categories of
highest weight representations of semisimple Lie algebras [Dix77], [Duf77],
[Jos79], [BG80], and thereafter with Hecke categories and Soergel bimodules
[BB81], [Soe92], [BG99].

The representation theory of affine Lie algebras is by now a mature subject,
with well understood theories of highest weight representations, affine Hecke
categories, and affine Soergel bimodules. However, a corresponding theory of
affine Harish-Chandra bimodules has been elusive. This absence has been felt
by multiple groups of experts for at least a quarter century, notably in the
work and conjectures of I. Frenkel and Malikov [FM97], [FM99].

### 1.2.

In this paper we introduce such a category of affine Harish-Chandra bimodules,
and in the process exhibit a partial substitute for infinitesimal characters
for Kac–Moody Lie algebras. To show our category of bimodules has the sought-
for properties, we provide a new type of singular localization onto Whittaker
flag manifolds. Our method of proof for localization, which uses techniques
from categorical representation theory, lifts the fundamental identification
of the Grothendieck group of a block of Category $\mathscr{O}$ with an induced
representation of the Hecke algebra to an equivalence of the block itself with
an induced representation of the Hecke category.

In the remainder of the introduction, we describe this all in more detail;
after giving the definition of the category of bimodules considered in Section
1.3, we then discuss our main Theorems 1.4.2 and 1.5.2.

###  1.3. Affine Harish-Chandra bimodules

#### 1.3.1.

There are two basic difficulties in giving an analogue of Harish-Chandra
bimodules with generalized infinitesimal characters in the affine setting. We
now review these, and explain the approach taken in the present work to
addressing them.

#### 1.3.2.

The first problem is to give a good notion of a general affine Harish-Chandra
bimodule. To see the difficulty, fix an almost simple, simply connected
algebraic group $G$ with Lie algebra $\mathfrak{g}$. Recall that a Harish-
Chandra bimodule is a $\mathfrak{g}\oplus\mathfrak{g}$-module on which the
diagonal action of $\mathfrak{g}$ integrates to an action of $G$.

Let us now pass to the loop group $G(\\!(z)\\!)$, whose Lie algebra is the
loop algebra $\mathfrak{g}(\\!(z)\\!)$. One may again consider the abelian
category of smooth
$\mathfrak{g}(\\!(z)\\!)\oplus\mathfrak{g}(\\!(z)\\!)$-modules on which the
diagonal action is integrable. However, as $G(\\!(z)\\!)$ has no nontrivial
smooth representations, a short argument shows that this implies that all of
$\mathfrak{g}(\\!(z)\\!)\oplus\mathfrak{g}(\\!(z)\\!)$ acts trivially. So the
obtained category is equivalent to vector spaces, and is too small to be of
use.

When one passes from the loop algebra to an affine Lie algebra
$\widehat{\mathfrak{g}}_{\kappa}$, i.e. one introduces the central charge of
level $\kappa$, problems of a similar nature arise.

#### 1.3.3.

To obtain more objects, we follow a proposal of Gaitsgory [Gai07]. Namely, if
one tries to produce more objects by passing to the derived category of
bimodules, one encounters the same degeneracy as in the abelian case. However,
E. Frenkel and Gaitsgory introduced in [FG09a] a certain enlargment of the
derived category, which we denote below by

(1.1)
$\mathfrak{g}(\\!(z)\\!)\oplus\mathfrak{g}(\\!(z)\\!)\operatorname{-mod}.$

This dg-category, sometimes called the renormalized derived category, contains
the usual unbounded derived category as a full subcategory, and carries a
$t$-structure for which the bounded below objects agree with the usual bounded
below derived category. Said informally, one adjoins objects concentrated in
cohomological degree $-\infty$. The role of such objects and such a category
had been long anticipated in the rich folklore around semi-infinite
mathematics, with important contributions by Arkhipov, Feigin, Finkelberg, E.
Frenkel, Positselski, Voronov, and others; see [Vor93], [Pos10] for a partial
indication.111As a striking example, the reader may wish to compare Voronov’s
final remarks in [Vor93] with Krause’s renormalization of the category of
quasi-coherent sheaves on a Noetherian scheme [Kra05], i.e. the category of
ind-coherent sheaves [Gai13].

One then takes the category of affine Harish-Chandra bimodules to be the
$G(\\!(z)\\!)$-equivariant objects in the enlarged category (1.1). To make
sense of this requires some categorical representation theory of groups.
Briefly, the diagonal adjoint action of $G(\\!(z)\\!)$ on
$\mathfrak{g}(\\!(z)\\!)\oplus\mathfrak{g}(\\!(z)\\!)$ induces an action of
the category of D-modules $D(G(\\!(z)\\!))$, viewed as a monoidal dg-category
under convolution, on (1.1). A general procedure then attaches to any
cocomplete dg-category $\mathscr{C}$ equipped with an action of
$D(G(\\!(z)\\!))$ its category of equivariant objects. If one runs the above
instead for the finite dimensional group $G$, this recovers the usual
(derived) category of Harish-Chandra bimodules.

#### 1.3.4.

At the time of Gaitsgory’s proposal, the formalism for passing to equivariant
objects was not yet available. Plainly, one would like to compute such a
category as the totalization of a cobar resolution akin to the simplicial
presentation of the equivariant derived category of sheaves on a quotient
stack. However, this uses inverse limits of categories, which cannot be
sensibly defined working at the triangulated level. The requisite categorical
representation theory has been developed over the ensuing fifteen years by
Gaitsgory, with significant contributions by Beraldo, Ben-Zvi–Nadler, and
Raskin, and building on earlier work of Beilinson and Drinfeld [BD], [FG06],
[BZN09], [Ber17], [Ras21], [Ras20], [Gai20].

The above category of affine Harish-Chandra bimodules, and its analogues at
other central charges, figures prominently in the main conjectures of the
local quantum Langlands correspondence, cf. [Gai07], [Gai18]. To our knowledge
the present work is the first to undertake its study.

#### 1.3.5.

We now turn to the second basic difficulty. Recall that in finite type to
obtain categories of Harish-Chandra bimodules equivalent to Hecke categories
and categories of highest weight representations, one fixes a strict or
generalized infinitesimal character for the underlying
$\mathfrak{g}\oplus\mathfrak{g}$-module.

This forces us to confront a fundamental issue in the representation theory of
affine algebras, which experts have contended with since the early days of the
subject. Namely, the center of the completed enveloping algebra $U(\hskip
1.42262pt\widehat{\mathfrak{g}}_{\kappa})$ is trivial for all levels $\kappa$
excepting the critical level $\kappa_{c}$. Nonetheless, it has long been
apparent that its category of modules behaves in certain respects as though
there is a nontrivial center. Indeed, in the literature one finds various
categories of representations whose linkage principles all take the same form;
see [DGK82], [LZ93], [Yak07], [ALZ16]. Namely, each time at a noncritical
level $\kappa$ they are indexed by orbits in the dual abstract Cartan
$\mathfrak{t}^{*}$ of the level $\kappa$ dot action of the affine Weyl group.

So, we must look for an alternative way to isolate, for any noncritical level
$\kappa$ and affine Weyl group orbit $\chi$, a full subcategory of modules

(1.2)
$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}_{\chi}\subset\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod},$

which acts as though it were the category of modules with ‘generalized
infinitesimal character $\chi$’.

#### 1.3.6.

Our approach to this second problem, which is perhaps the basic new idea of
the present work, is again via categorical representation theory.

Recall that, given an algebra $A$ and a module $M$, from any element $m$ of
$M$ one can form the $A$-submodule generated by $m$. Similarly, given a
monoidal category and a module category, to any object of the latter one can
associate the submodule generated by it.

In this way, we define (1.2) to be the $D_{\kappa}(G(\\!(z)\\!))$-submodule of
$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}$ generated by the Verma
module of highest weight $\lambda$, for any element $\lambda$ of the orbit
$\chi$; the resulting submodule is independent of the choice. If one runs this
definition for the finite dimensional group $G$, it recovers the usual derived
category of $\mathfrak{g}$-modules with generalized infinitesimal character
that of the corresponding Verma module. So, we use the categorical action to
bypass the absence of infinitesimal characters.

Combining this and the construction (1.1), we obtain for appropriate pairs of
weights $\chi,\phi$ and levels $\kappa,\kappa_{\circ}$ the desired categories
of affine Harish-Chandra bimodules.222The term ‘appropriate’ here means that,
just as in finite type, the difference between $(\chi,\kappa)$ and
$(\phi,\kappa_{\circ})$ should be integral (or the categories would vanish).
Namely, we again pass to $G(\\!(z)\\!)$-equivariant objects to obtain the
submodule

(1.3)
$\operatorname{HCh}_{\chi,\phi}:=(\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}_{\chi}\otimes\hskip
1.42262pt\widehat{\mathfrak{g}}_{\kappa_{\circ}}\operatorname{-mod}_{\phi})^{G(\\!(z)\\!)}\subset\widehat{\mathfrak{g}}_{\kappa}\oplus\widehat{\mathfrak{g}}_{\kappa_{\circ}}\operatorname{-mod}^{G(\\!(z)\\!)}.$

#### 1.3.7.

We spend the majority of the paper verifying that the categories of Harish-
Chandra bimodules (1.3) introduced above behave in the desired ways. This is
not completely transparent, since in the most important cases every single
object of $\operatorname{HCh}_{\chi,\phi}$ is a homological phantom, i.e. is
concentrated in cohomological degree $-\infty$.

#### 1.3.8.

Before discussing the bimodules, let us briefly describe some features of the
categories of $\widehat{\mathfrak{g}}_{\kappa}$-modules appearing in (1.2). As
$\chi$ runs through the set of affine Weyl group orbits, the subcategories
(1.2) contain all parabolic inductions of modules with infinitesimal
characters from proper Levi factors of $\widehat{\mathfrak{g}}_{\kappa}$.
Moreover, they satisfy the linkage principle

(1.4) $\underset{\chi}{\oplus}\hskip
2.84526pt\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}_{\chi}\subset\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod},$

i.e. for distinct orbits $\chi_{1},\chi_{2}$ the categories
$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}_{\chi_{1}}$ and
$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}_{\chi_{2}}$ are left and
right orthogonal to one another. Combining these points confirms a linkage
principle for positive energy representations predicted by Yakimov [Yak07],
cf. Section 8 for more details.

If $\kappa$ is an integral level, under the (conjectural) local geometric
Langlands correspondence, one can associate to
$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}$ a sheaf of categories
$\mathscr{S}$ on the moduli space of $\check{G}$-local systems on the
punctured disk, where $\check{G}$ denotes the Langlands dual of $G$. The
inclusion (1.4) should correspond to the subcategories of $\mathscr{S}$
supported on formal neighborhoods of local systems with regular singularities.
In particular, the inclusion (1.4) does not see the wildly ramified part of
$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}$. However, for our
purposes, namely to have the sought-for relations with affine Hecke
categories, this is a necessary feature, and not a bug. We turn to these
relations next.

###  1.4. Equivalences with affine Hecke categories

#### 1.4.1.

In the next theorem we identify our categories of bimodules (1.3) with
variants of the affine Hecke category under an integrality assumption. This
result strongly suggests that our definition is the correct one.

To state it, we need some notation. Let $\kappa,\kappa_{\circ}$ be noncritical
integral levels. At such levels, the dot action of the affine Weyl group
preserves the weight lattice. Denote by $\chi$ and $\phi$ a pair of orbits of
weights at levels $\kappa$ and $\kappa_{\circ}$, respectively. Fix a pinning
of the loop group, and in particular an Iwahori subgroup

$I\subset G(\\!(z)\\!),$

whose prounipotent radical we denote by $\mathring{I}\subset I$. By a standard
procedure, one may then associate to $\chi$ and $\phi$ character sheaves on
$\mathring{I}$, which we denote by $\psi$ and $\omega$, respectively. Loosely,
as the orbit $\chi$ grows more singular the character $\psi$ grows more
nondegenerate, cf. Section 7.1 for the details.

We also need one more definition, which may be new. For a character sheaf
$\psi$ as above, consider the corresponding category of Iwahori–Whittaker
D-modules on the loop group

(1.5) $D\big{(}G(\\!(z)\\!)/\mathring{I},-\psi\big{)}.$

This is a naturally a $D(G(\\!(z)\\!))$-module under left convolution. We may
therefore consider the submodule generated by the Iwahori–equivariant objects.
This is a categorical analogue of the (degenerate) Steinberg module of a
finite group of Lie type, and is a colocalization of (1.5), i.e. its inclusion
admits a right adjoint.

Since (1.5) corepresents the functor of taking the Iwahori–Whittaker
equivariant objects $\mathscr{C}^{\mathring{I},\psi}$ of a categorical
representation $\mathscr{C}$ of the loop group, its Steinberg submodule
determines a corresponding full subcategory of Steinberg–Whittaker equivariant
objects

$\mathscr{C}^{\mathring{I},\psi,\operatorname{s}}\subset\mathscr{C}^{\mathring{I},\psi}.$

For example, if $\psi$ is the trivial character sheaf, the Steinberg–Whittaker
invariants are the Iwahori monodromic objects of $\mathscr{C}$. Finally, we
note that the Steinberg–Whittaker invariants are also introduced in
forthcoming work of Arinkin and Bezrukavnikov.

With these preparations, we may state our basic theorem on affine Harish-
Chandra bimodules.

######  Theorem 1.4.2.

Let $\chi,\phi$ and $\psi,\omega$ be as above. There are canonical
equivalences

$\operatorname{HCh}_{\chi,\phi}\simeq D\big{(}\hskip
0.85358pt(\mathring{I},\psi,\operatorname{s})\backslash\hskip
0.28453ptG(\\!(z)\\!)\hskip 0.28453pt/\hskip
0.85358pt(\mathring{I},-\omega,\operatorname{s})\hskip
0.85358pt\big{)}\simeq\widehat{\mathfrak{g}}_{\kappa_{\circ}}\operatorname{-mod}_{\phi}^{\mathring{I},\psi}.$

We now make a few remarks about Theorem 1.4.2.

######  Remark 1.4.3.

The analogues of the above equivalences for the finite dimensional group $G$
are fundamental results, due in various amounts of generality to
Bernstein–Gelfand, Beilinson–Bernstein, Milicic–Soergel, and more recently
Ginzburg, Webster, and Losev [BG80], [BB81], [MS97], [Gin09], [Web11],
[Los12]. However, as far as we are aware, in the case of $\chi$ and $\phi$
both singular, it was only a folklore expectation that there should be a
realization as appropriate bi-Whittaker sheaves.

For context, we note that the analogous categories of bi–Whittaker sheaves of
general Kac–Moody groups should be Koszul dual to bi-parabolic Hecke
categories, and admit equivalences with singular Soergel bimodules.

We should also mention that in the sheaf–theoretic realization, the Steinberg
condition on the Whittaker invariants appearing on one side in fact implies it
holds on the other. For this reason, in the literature one does not find the
Steinberg–Whittaker invariants explicitly introduced except in the case of the
trivial character.

Finally, we emphasize that, to us, the main point of Theorem 1.4.2 is that our
category of affine Harish-Chandra bimodules is really the sought-for one,
i.e., it behaves like its finite dimensional counterpart. Unfortunately, to
prove that this is so, we cannot mimic the existing arguments. Namely, much of
the above work relies on favorable properties of Harish-Chandra bimodules with
infinitesimal characters. We only obtain their affine analogues a posteriori
as consequences of the theorem.

######  Remark 1.4.4.

The bimodule and sheaf-theoretic categories appearing above have natural
convolution structures. Namely, given affine Harish-Chandra bimodules
$\mathscr{M}_{1}$ and $\mathscr{M}_{2}$ at appropriate levels, one may compose
them by forming the semi-infinite homology

$\mathscr{M}_{1}\underset{\widehat{\mathfrak{g}}}{\overset{\frac{\infty}{2}}{\otimes}}\hskip
1.42262pt\mathscr{M}_{2}.$

By construction the equivalence of Theorem 1.4.2 exchanges this with the
convolution of the corresponding D-modules on the loop group, cf. Remark
10.4.2 for the details.

######  Remark 1.4.5.

We also introduce the analogues of categories with strict infinitesimal
characters. To see how, recall that in finite type, the category
$\mathfrak{g}\operatorname{-mod}_{\lambda}$ is acted on by a completion of
$Z(\mathfrak{g})$, i.e. the log of monodromy. We extend this to affine type by
recognizing this as the equivariant Hochschild cohomology of
$\mathfrak{g}\operatorname{-mod}_{\lambda}$. We may then set the monodromy to
zero, cf. Section 9 for the details.

###  1.5. Theorem 1.4.2 via Steinberg–Whittaker localization

#### 1.5.1.

The idea of the proof of Theorem 1.4.2 is simple. First, some abstract
nonsense: if $\mathscr{C}$ and $\mathscr{D}$ are categorical loop group
representations, such that $\mathscr{C}$ is dualizable as an abstract
cocomplete dg-category, i.e. as an object of
$\operatorname{DGCat}_{\operatorname{cont}}$, we have canonical
identifications

(1.6)
$\operatorname{Hom}_{D(G(\\!(z)\\!))\operatorname{-mod}}(\mathscr{C},\mathscr{D})\simeq\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\mathscr{C},\mathscr{D})^{G(\\!(z)\\!)}\simeq(\mathscr{C}^{\vee}\otimes\mathscr{D})^{G(\\!(z)\\!)},$

cf. Section 2 below for the precise definitions of
$D(G(\\!(z)\\!))\operatorname{-mod}$ and
$\operatorname{DGCat}_{\operatorname{cont}}$.

We apply this as follows. For Kac–Moody representations, the operation of
taking semi-infinite cohomology yields a canonical duality

(1.7)
$\displaystyle\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{\vee}\simeq\widehat{\mathfrak{g}}_{-\kappa+2\kappa_{c}}\operatorname{-mod},$
where as before $\kappa_{c}$ denotes the critical level. Writing $\rho$ for
the half sum of the positive roots of $\mathfrak{g}$, this restricts, for any
$\lambda\in\mathfrak{t}^{*}$, to a duality (1.8)
$\displaystyle\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}_{\lambda}^{\vee}\simeq\widehat{\mathfrak{g}}_{-\kappa+2\kappa_{c}}\operatorname{-mod}_{-\lambda-2\rho}.$

In particular, by combining (1.6) and (1.8), we recognize our categories of
Harish-Chandra bimodules as spaces of intertwining operators between
categorical representations.

With this, Theorem 1.4.2 follows formally from rewriting our categories of
Kac–Moody representations as categories of D-modules. We do so in the
following theorem, which we call Steinberg–Whittaker localization.

######  Theorem 1.5.2.

Let $\chi,\kappa,$ and $\psi$ be as in Theorem 1.4.2. Then there is a
canonical $D(G(\\!(z)\\!))$-equivariant equivalence

$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}_{\chi}\simeq
D\big{(}G(\\!(z)\\!)/\mathring{I},\psi,\operatorname{s}\big{)}.$

Let us discuss the contents of Theorem 1.5.2 in several cases.

#### 1.5.3.

If $\chi$ is regular of negative level, then $\psi$ is trivial, and Theorem
1.5.2 recovers the usual Kashiwara–Tanisaki localization theorem on the
enhanced affine flag variety [KT95].

#### 1.5.4.

By contrast, if $\chi$ is singular of negative level, to our knowledge the
theorem and its analogue for a reductive group $G$ are both new, and are one
of the main results of the present work.

Let us focus on the maximally singular case, e.g. for $G$ with $\chi=-\rho$.
Here, the theorem says that localization onto the nondegenerate Whittaker flag
manifold yields an equivalence

(1.9) $\mathfrak{g}\operatorname{-mod}_{-\rho}\simeq
D\big{(}G/N,\psi,\operatorname{s}).$

To get a feel for what is going on, recall that in the regular case, i.e.
Beilinson–Bernstein localization, the underlying quasicoherent sheaves of the
obtained D-modules on $G/N$ descend to the flag variety $G/B$. This
homogeneous space parametrizes maximal unipotent subalgebras
$\mathfrak{n}\subset\mathfrak{g}$, and the fibres of the obtained sheaf store
(a generalized weight space in) the corresponding $\mathfrak{n}$-homology of
the initial $\mathfrak{g}$-module.

In the maximally singular case (1.9), the underlying quasicoherent sheaves of
the obtained D-modules on $G$ descend to $G/Z_{G}N$, where $Z_{G}$ denotes the
center of $G$. This homogeneous space parametrizes maximal unipotent
subalgebras $\mathfrak{n}$ equipped with a generic additive character $\Psi$,
and the fibres of the obtained sheaf store the corresponding $\Psi$-twisted
$\mathfrak{n}$-homology of the initial $\mathfrak{g}$-module.

Since the work of Beilinson–Bernstein, the problem of singular localization
has been studied by various authors to various ends, with important
contributions by Backelin–Kremnizer, Beilinson–Ginzburg,
Bezrukavnikov–Mirkovic–Rumynin, and Kashiwara [BK15], [BG99], [BMR06],
[Kas89].

To our knowledge (1.9) may be the first nontrivial identification of
$\mathfrak{g}\operatorname{-mod}_{-\rho}$ with a category of D-modules.
Indeed, for any singular generalized infinitesimal character, localization
onto the enhanced flag variety $G/N$ is not fully faithful. A fully faithful
localization functor for singular categories was introduced by
Backelin–Kremnizer in the interesting work [BK15]. As they note, however, for
maximally singular blocks it is a special case of the tautological equivalence
between all $\mathfrak{g}$-modules and all weakly $G$-equivariant D-modules on
$G$.

#### 1.5.5.

Finally, unlike at negative level, where one can show the equivalences are
$t$-exact, the equivalences for $\chi$ at positive level are of infinite
cohomological amplitude, and make essential use of the renormalization of the
underlying triangulated categories. They may be considered an affine analogue
of usual localization composed with the long intertwining operator.

###  1.6. Theorem 1.5.2 via categorical parabolic induction

Finally, we discuss the proof of Steinberg–Whittaker localization.

#### 1.6.1.

The basic idea is as follows. Many categories in representation theory are
well known to ‘look like’ induced representations at the level of Grothendieck
groups.

In the example at hand, blocks of Category $\mathscr{O}$ ‘look like’
antispherical modules for the Hecke category, i.e. inductions of sign
characters. This point of view has been emphasized by Soergel and his school,
and in particular plays a key role in the recent breakthroughs of Williamson
and others in the modular representation theory of reductive groups.

As we show in this article, the induction picture of the Grothendieck group
underlies an induction at the level of categories. Given this, the
localization theorems are easy consequences.

#### 1.6.2.

Let us formulate the contents of the preceding paragraph precisely. The
discussion below is for $G$, the case of the loop group being similar.

Fix an antidominant weight $\chi$, and write $P$ for the standard parabolic
corresponding the stabilizer of $\chi$ under the dot action of the Weyl group.
Write $N_{P}$ for the unipotent radical of $P$, and $L$ for its Levi quotient,
with corresponding Lie algebra $\mathfrak{l}$. Note that $\chi$ is a maximally
singular weight for $L$.

Consider the functor of parabolic induction of Lie algebra representations
$\operatorname{pind}:\mathfrak{l}\operatorname{-mod}_{\chi}\rightarrow\mathfrak{g}\operatorname{-mod}_{\chi}^{N_{P}}.$
By adjunction, this induces a functor

(1.10)
$D(G)\underset{D(P)}{\otimes}\mathfrak{l}\operatorname{-mod}_{\chi}\rightarrow\mathfrak{g}\operatorname{-mod}_{\chi},$

and we show in Theorem 7.2.5 that this is an equivalence.

######  Remark 1.6.3.

On $B$-equivariant objects, (1.10) yields a canonical equivalence

$D(B\backslash G/B)\underset{D(B\backslash
P/B)}{\otimes}\mathfrak{l}\operatorname{-mod}_{\chi}^{B}\simeq\mathfrak{g}\operatorname{-mod}_{\chi}^{B}.$

That is, blocks of Category $\mathscr{O}$ are inductions of sign
representations at the level of categories. Similarly, in geometry we find
that convolution induces an equivalence

$D(B\backslash G/B)\underset{D(B\backslash P/B)}{\otimes}D(B\backslash
P/N,\psi)\simeq D(B\backslash G/N,\psi),$

where $\psi$ is a generic additive character of $N/N_{P}$.

#### 1.6.4.

Let us turn to localization theory. In the maximally singular case (1.9),
after introducing a functor between the two sides, by abstract nonsense it
suffices to check the obtained map on $B$-equivariant categories is an
equivalence. In this case, both sides are equivalent to the category of vector
spaces, so this is an easy check. The general case then follows formally by
parabolically inducing (1.9). A similar argument also recovers the singular
localization of Backelin–Kremnizer.

######  Remark 1.6.5.

In the case of regular $\chi$, one can ask how this treatment of localization
compares to earlier ones. The construction of the functors is somewhat
different, especially so in affine type. As for the proof, the reduction to an
equivalence on Borel-equivariant objects first appeared in work of E.
Frenkel–Gaitsgory [FG09b]. From there, they rely on an explicit nontrivial
calculation by Kashiwara–Tanisaki, while we note it follows directly from the
compatibility with the Hecke action.

######  Remark 1.6.6.

One can also ask about the strict logical dependence of the arguments in the
present paper on prior results.333Of course, the non-logical dependence on
earlier work is immense, and difficult to overstate. Aside from facts in
categorical representation theory (of the flavor of abstract nonsense), our
proofs of the main Theorems 1.4.2 and 1.5.2 are self-contained except for some
basic facts about the Schubert decomposition of flag varieties and Category
$\mathscr{O}$ for $\mathfrak{sl}_{2}$.

After this, we do use two nontrivial results from representation theory. To
obtain the block decompositions for our categories of
$\widehat{\mathfrak{g}}_{\kappa}$-modules and bimodules, we use Kac–Kazhdan’s
calculation of the Shapovalov determinant [KK79]. In addition, to obtain the
linkage principle for positive energy representations predicted by Yakimov, we
need to identify our construction of $\mathfrak{g}\operatorname{-mod}_{\chi}$
with the usual description via infinitesimal characters. This is deduced from
Beilinson–Bernstein’s calculation of global twisted differential operators on
the finite flag manifold [BB81].

#### 1.6.7.

We would like to finish with a couple comments which lie outside the scope of
the article.

First, the presented arguments adapt, mutatis mutandis, to the case of
Kac–Moody groups and blocks containing a simple Verma module with finite Weyl
group stabilizer (or the dual of such a block). It would be very interesting
to find a similar approach to localization in positive characteristic and for
quantum groups.

Second, while we address the question raised by I. Frenkel–Malikov in [FM97]
and [FM99] of finding the desired category of affine Harish-Chandra bimodules,
we do not address some striking conjectures of loc. cit. on the relation with
Kazhdan–Lusztig fusion. There indeed is a monoidal functor of the anticipated
form from the Kazhdan–Lusztig category to the present category of bimodules.
This, along with the closely related theory of projective functors in affine
type and principal series Harish-Chandra bimodules, will be discussed in a
sequel to the present work.

Third, we would like to mention that forthcoming work of Losev produces
categories of Harish-Chandra bimodules for quantum groups. We expect that one
has equivalences between these and the categories considered in this paper,
compatible with the Kazhdan–Lusztig equivalence.

###  1.7. Organization of the paper

Throughout the first half of the paper, we alternate between generalities in
categorical representation theory, and their applications to the problem at
hand.

In Section 2, we gather preliminary material and set notation.

In Section 3, we study the basic adjunctions between representations of Hecke
categories and groups. This is then used in Section 4, where we introduce the
categories of Lie algebra representations studied in this paper.

In Section 5, we prove the localization theorems in the maximally singular
case. We then study the functor of parabolic induction for categorical
representations, and crucially its relation with induction for Hecke
categories, in Section 6. This is used in Section 7 to deduce the general case
of the localization theorems.

The remaining sections then apply the previous material to various problems in
representation theory, and may be read independently of one another. In
Section 8, we obtain a linkage principle for positive energy representations
of affine Lie algebras suggested by Yakimov. In Section 9, we introduce
equivariant Hochschild cohomology and the analogues of strict infinitesimal
characters. Finally, in Section 10, we obtain the main theorems on affine
Harish-Chandra bimodules.

Acknowledgements. This paper arises from pairing methods from categorical
representation theory with perspectives on Hecke categories and their modules
we learned from the work of Soergel and his school.

For the former subject, we are very grateful to Dennis Gaitsgory, Sam Raskin,
and David Yang for stimulating conversations. In particular, we thank Sam
Raskin for sharing with us his approach to Kashiwara–Tanisaki localization via
convolution with a Verma module, which greatly influenced our thinking.

For the latter subject, we thank Roman Bezrukavnikov, Ben Elias, Ivan Losev,
Shotaro Makisumi, Ben Webster, Geordie Williamson, and Zhiwei Yun for many
helpful conversations.

We would also like to thank Levent Alpoge, Tony Feng, Sasha Goncharov, David
Jordan, Apoorva Khare, Fedor Malikov, Wolfgang Soergel, and Milen Yakimov for
stimulating conversations and correspondence related to the present work.

Finally, particular thanks are due to Igor Frenkel, who shared with us some of
his beautiful expectations about affine Harish-Chandra bimodules and principal
series for loop groups several years ago.

##  2\. Preliminaries and notation

In this section we collect some background material and set notation. The
reader may wish to, after possibly glancing at Sections 2.2.1 and 2.5.3 for
what we denote by $G$, $B$, and $\mathfrak{g}\operatorname{-mod}$, move
directly to Section 3 and refer back only as needed.

###  2.1. Differential graded categories

A reference for the following material is Chapter 1 of [GR17].

#### 2.1.1.

Throughout the paper, we work over a fixed algebraically closed field $k$ of
characteristic zero. We denote by $\operatorname{DGCat}_{\operatorname{cont}}$
the $(\infty,2)$-category of cocomplete, presentable $k$-linear dg-categories.
Unless otherwise specified, by a category $\mathscr{C}$ we mean an object of
$\operatorname{DGCat}_{\operatorname{cont}}$. By a functor between categories
$\mathscr{C}\rightarrow\mathscr{D}$ we mean an object of the corresponding
category444We view it as a dg-category, using the fact that
$\operatorname{DGCat}_{\operatorname{cont}}$ is enriched over itself. It is
convenient to express this dg-category as an internal Hom using the Lurie
tensor product discussed below. In doing so, one can avoid the theory of
$(\infty,2)$-categories entirely if desired. of maps
$\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\mathscr{C},\mathscr{D}),$
i.e. a continuous quasi-functor; see for example [Dri04] for a review of
quasi-functors.

Throughout the paper, when we consider a limit or colimit of categories, we
mean the (co)limit taken in $\operatorname{DGCat}_{\operatorname{cont}}$.

#### 2.1.2.

We recall that $\operatorname{DGCat}_{\operatorname{cont}}$ carries the
structure of a unital symmetric monoidal $\infty$-category with respect to the
Lurie tensor product. The unit is given by $\operatorname{Vect}$, the category
of $k$-vector spaces, and we denote the underlying binary product of
categories $\mathscr{C}$ and $\mathscr{D}$ by $\mathscr{C}\otimes\mathscr{D}$.
Given objects $c$ of $\mathscr{C}$ and $d$ of $\mathscr{D}$, we denote the
corresponding object of $\mathscr{C}\otimes\mathscr{D}$ by $c\boxtimes d$.

If a category $\mathscr{C}$ is dualizable with respect to the Lurie tensor
product, we denote its dual by $\mathscr{C}^{\vee}$. Plainly, this is
equivalent to, for any category $\mathscr{S}$, the natural map

$\displaystyle\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\mathscr{C},\operatorname{Vect})\otimes\mathscr{S}$
$\displaystyle\simeq\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\mathscr{C},\operatorname{Vect})\otimes\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\operatorname{Vect},\mathscr{S})$
$\displaystyle\rightarrow\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\mathscr{C},\mathscr{S})$

being an equivalence (or in fact, just for $\mathscr{S}=\mathscr{C}$), in
which case
$\mathscr{C}^{\vee}\simeq\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\mathscr{C},\operatorname{Vect})$.

Of particular importance to us is the resulting identification of the category
of endo-functors of a dualizable category $\mathscr{C}$

(2.1)
$\mathscr{C}^{\vee}\otimes\mathscr{C}\simeq\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\mathscr{C},\mathscr{C}).$

This is an equivalence of monoidal $\infty$-categories, where on the left the
underlying binary product is induced by contraction of tensors using the
evaluation pairing

$\operatorname{ev}:\mathscr{C}\otimes\mathscr{C}^{\vee}\rightarrow\operatorname{Vect}.$

I.e., it is given by the composition

$\mathscr{C}^{\vee}\otimes\mathscr{C}\otimes\mathscr{C}^{\vee}\otimes\mathscr{C}\xrightarrow{\operatorname{id}_{\mathscr{C}^{\vee}}\otimes\operatorname{ev}\otimes\operatorname{id}_{\mathscr{C}}}\mathscr{C}^{\vee}\otimes\operatorname{Vect}\otimes\hskip
1.42262pt\mathscr{C}\simeq\mathscr{C}^{\vee}\otimes\mathscr{C}.$

Finally, if $\mathscr{C}$ and $\mathscr{D}$ are dualizable, one then has a
canonical equivalence

$\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\mathscr{C},\mathscr{D})\simeq\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\mathscr{D}^{\vee},\mathscr{C}^{\vee}),$

and in particular for a functor $F:\mathscr{C}\rightarrow\mathscr{D}$ one has
a dual functor $F^{\vee}:\mathscr{D}^{\vee}\rightarrow\mathscr{C}^{\vee}$.

###  2.2. Groups and Lie algebras

#### 2.2.1.

Throughout the paper, we use the letter $G$ to denote either (i) a connected
reductive group over $k$, or (ii) the loop group of an almost simple, simply
connected group $H$.555One may deduce from case (ii) similar results, mutatis
mutandis, for (the neutral component of) the loop group of a connected
reductive group. We recall that the latter is an ind-scheme of infinite type
over $k$.

We use the letter $B$ to denote in case (i) a Borel subgroup of $G$, or in
case (ii) an Iwahori subgroup. By the latter, we mean the preimage in the arc
group of $H$ of a Borel subgroup of $H$. We denote the prounipotent radical of
$B$ by $N$, and write $T$ for the abstract Cartan $B/N$. We denote the
character and cocharacter lattices of $T$ by $\Lambda$ and $\check{\Lambda}$,
respectively, in case (i), and $\Lambda_{f}$ and $\check{\Lambda}_{f}$,
respectively, in case (ii). The subscript ‘$f$’ is for ‘finite’; we will
explain what we denote by $\Lambda$ in case (ii) below in Section 2.5.

#### 2.2.2.

We denote by $\mathfrak{g}$ the Lie algebra of $G$ in case (i), and the
topological Lie algebra in case (ii). Explicitly, if in the latter case we
write $\mathfrak{h}$ for the Lie algebra of $H$, we have

$\mathfrak{g}\simeq\mathfrak{h}\underset{k}{\otimes}k(\\!(z)\\!).$

#### 2.2.3.

Given a level $\kappa$ for $\mathfrak{h}$, i.e. an
$\operatorname{Ad}$-invariant symmetric bilinear form on $\mathfrak{h}$, we
denote the corresponding Kac–Moody extension by

(2.2) $0\rightarrow
k\cdot\mathbf{1}\rightarrow\widehat{\mathfrak{h}}_{\kappa}\rightarrow\mathfrak{h}\underset{k}{\otimes}k(\\!(z)\\!)\rightarrow
0.$

Recall there exists a basic form $\kappa_{b}$, which gives the short coroots
squared length two. We call a form $\kappa$ integral if it is an integral
multiple of $\kappa_{b}$.

Recall that the critical level $\kappa_{c}$ is by definition minus one half
times the Killing form. We say a noncritical level $\kappa$ is positive if

$\kappa-\kappa_{c}\hskip 1.42262pt\in\hskip
1.42262pt\mathbb{Q}^{>0}\cdot\kappa_{b},$

and otherwise call it negative.

###  2.3. D-modules

Some references for the following material are [Ras15] and the very readable
[Ber17].

#### 2.3.1.

For a quasi-compact, quasi-separated scheme $X$ of finite type over $k$, we
denote by $D(X)$ the canonical dg-enhancement of its unbounded derived
category of D-modules.

We will also need categories of D-modules on certain infinite dimensional
varieties, which we again denote by $D(X)$; this is denoted by $D^{*}(X)$ in
the above references.

For an ind-scheme of ind-finite type, e.g. the affine flag variety, given a
presentation as a filtered colimit of varieties of finite type along closed
embeddings $X=\varinjlim_{\alpha}X_{\alpha},$ its category of D-modules is the
colimit (in $\operatorname{DGCat}_{\operatorname{cont}}$)

$D(X)\simeq\varinjlim_{\alpha}D(X_{\alpha}).$

Plainly, $D(X)$ is compactly generated by $*$-pushforwards of coherent
complexes of D-modules from all $X_{\alpha}$, with the natural chain complexes
of homomorphisms between such pushforwards.

We note that $D(X)$, like many categories met in this paper, carries a natural
$t$-structure but is not in general (1) the derived category of its heart, nor
(2) left complete, i.e. it may contain objects in degrees $\leqslant n$ for
all integers $n$ with respect to the $t$-structure.

Recall that a scheme $Y$ is called _placid_ if it admits a presentation
$Y=\varprojlim_{\beta}Y_{\beta}$ as a cofiltered limit of varieties under
affine smooth maps (e.g. the Iwahori subgroup). Its category of D-modules is
the colimit

$D(Y)\simeq\varinjlim_{\beta}D(Y_{\beta}).$

Plainly, $D(Y)$ is compactly generated by $!$-pullbacks of coherent complexes
from all $Y_{\beta}$, with homomorphisms the colimit of the complexes of
homomorphisms of the pullbacks to $Y_{\beta^{\prime}}$, for all sufficiently
large $\beta^{\prime}$.

Finally, a placid ind-scheme, e.g. the loop group, is an ind-scheme which
admits a presentation as a filtered colimit of placid schemes along closed
embeddings of finite presentation $Z=\varinjlim_{\gamma}Z_{\gamma}$. Its
category of D-modules is again the colimit of $*$-pushforwards

$D(Z)\simeq\varinjlim_{\gamma}D(Z_{\gamma}).$

Given a map $f:X\rightarrow Y$ of placid ind-schemes, one has an associated
$*$-pushforward

$f_{*}:D(X)\rightarrow D(Y).$

#### 2.3.2.

Finally, we should note such categories of D-modules are typically self-dual
as dg-categories. For a variety $X$, the duality pairing $D(X)\otimes
D(X)\rightarrow\operatorname{Vect}$ is given by

$M\boxtimes N\mapsto\Gamma_{\operatorname{dR}}(X,M\overset{!}{\otimes}N).$

For a map of varieties $X\rightarrow Y$, the functors of $*$-pushforward and
$!$-pullback are then dual.

Similarly, for a placid scheme, or ind-scheme of ind-finite type, its category
of D-modules is canonically self-dual. For a placid ind-scheme, a choice of
dimension theory, if it exists, yields a self-duality, cf. [Ras15].

###  2.4. Categorical representations

Some references for this material are [Ber17], [Ras20], and [Gai20].

#### 2.4.1.

If $K$ is a group placid ind-scheme, its category of D-modules $D(K)$ is
naturally an algebra object in $\operatorname{DGCat}_{\operatorname{cont}}$,
whose underlying binary product is given by convolution

$D(K)\otimes D(K)\simeq D(K\times
K)\xrightarrow{\operatorname{mult}_{*}}D(K).$

One then has its associated $(\infty,2)$-category of left modules
$D(K)\operatorname{-mod}$. In particular, an object of
$D(K)\operatorname{-mod}$ is a category $\mathscr{C}$ equipped with an action
$D(K)\otimes\mathscr{C}\rightarrow\mathscr{C}$ which is associative and unital
up to coherent homotopy.

Inversion on $K$ yields a canonical equivalence between $D(K)$ and its
opposite algebra. This induces an identification of its categories of left and
right modules. In particular, for any $K$-modules $\mathscr{C}$ and
$\mathscr{D}$ the category of maps
$\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\mathscr{C},\mathscr{D})$
is naturally a $K\times K$-module.

As with modules for any algebra object $\mathscr{M}$ in
$\operatorname{DGCat}_{\operatorname{cont}}$, one has a conservative
‘forgetful’ functor

$\operatorname{Oblv}:\operatorname{Hom}_{D(K)\operatorname{-mod}}(\mathscr{C},\mathscr{D})\rightarrow\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\mathscr{C},\mathscr{D}).$

So, a $D(K)$-equivariant functor $F:\mathscr{C}\rightarrow\mathscr{D}$ is a
functor between the underlying categories equipped with a datum of
$D(K)$-equivariance.

We would like to record one useful property of $D(K)$ which is not enjoyed by
all algebras. Namely, any datum of lax or oplax equivariance on a functor
$F:\mathscr{C}\rightarrow\mathscr{D}$ of $D(K)$-modules is automatically
strict; this is Lemma D.4.4 of [Gai20]. In particular, any adjoint of a
$D(K$)-equivariant functor acquires a canonical datum of $D(K)$-equivariance.

#### 2.4.2.

An important class of categorical representations arises as follows. If $K$
acts on a placid ind-scheme $X$, this yields an action of $D(K)$ on $D(X)$.
Moreover, the duality pairing of Section 2.3.2 carries a canonical datum of
$D(K)$-equivariance.

In particular, taking $X$ to be a point, we obtain the trivial representation

$\operatorname{Vect}\in D(K\times K)\operatorname{-mod}.$

For any $D(K)$-module $\mathscr{C}$, we then have its associated invariants
and coinvariants, respectively given by

(2.3)
$\mathscr{C}^{K}:=\operatorname{Hom}_{D(K)\operatorname{-mod}}(\operatorname{Vect},\mathscr{C})\quad\text{and}\quad\mathscr{C}_{K}:=\operatorname{Vect}\underset{D(K)}{\otimes}\mathscr{C}.$

Explicitly, via the bar resolution, one has the semisimplicial presentation

(2.4) $\textstyle{\mathscr{C}^{K}\simeq\varprojlim\big{(}\hskip
1.42262pt\hskip
1.42262pt\mathscr{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hskip
1.42262pt\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(D(K),\mathscr{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hskip
1.42262pt\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(D(K)\otimes
D(K),\mathscr{C})\hskip 1.42262pt\cdots\hskip 1.42262pt\big{)}\hskip
1.42262pt,}$

where the arrows are induced as usual by action and projection; one also has a
similar semicosimplicial presentation of the coinvariants. There are
tautological ‘forgetting’ and ‘inserting’ $D(K)$-equivariant functors

$\operatorname{Oblv}:\mathscr{C}^{K}\rightarrow\mathscr{C}\quad\text{and}\quad\operatorname{ins}:\mathscr{C}\rightarrow\mathscr{C}_{K}.$

For $D(K)$-modules $\mathscr{C}$ and $\mathscr{D}$, passing from an
equivariant functor to the underlying functor between categories yields an
equivalence

$\operatorname{Hom}_{D(K)\operatorname{-mod}}(\mathscr{C},\mathscr{D})\simeq\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\mathscr{C},\mathscr{D})^{K},$

where on the right-hand side $K$ acts diagonally.

#### 2.4.3.

If $K$ is an affine group scheme whose prounipotent radical is of finite
codimension, then the tautological forgetful functor $\operatorname{Oblv}$
from invariants admits a right ‘averaging’ adjoint

$\operatorname{Oblv}:\mathscr{C}^{K}\rightleftarrows\mathscr{C}:\operatorname{Av}_{*}^{K}.$

Moreover, by adjunction $\operatorname{Av}_{*}^{K}$ factors as a map

$\mathscr{C}\xrightarrow{\operatorname{ins}}\mathscr{C}_{K}\rightarrow\mathscr{C}^{K},$

and the latter is an equivalence. That is, for ‘compact open’ groups $K$,
their invariants and coinvariants are canonically isomorphic, and in
particular commute with (co)limits and duality.

######  Example 2.4.4.

Suppose for simplicity that $K$ is of finite type, and acts on a variety $X$.
Then, by smooth descent for D-modules, one has a canonical equivalence

$D(X)^{K}\simeq D(X/K),$

where $D(X/K)$ denotes the category of D-modules on the stack quotient $X/K$,
cf. [BL94], [DG13] for descriptions of the latter. Writing $\pi:X\rightarrow
X/K$ for the projection, this exchanges $\operatorname{Oblv}$ with $\pi^{*}$,
and $\operatorname{Av}_{*}^{K}$ with $\pi_{*}$.

#### 2.4.5.

We will also need to consider invariants twisted by a character. So, suppose
$\chi$ is a character sheaf on $K$, i.e. a D-module invertible with respect to
the $!$-tensor product equipped with an isomorphism

$\operatorname{mult}^{!}\chi\simeq\chi\boxtimes\chi$

which is associative. One has a corresponding $D(K\times K)$-bimodule
structure on $\operatorname{Vect}$, which we denote by
$\operatorname{Vect}_{\chi}$, whose associated binary product is given by

$D(K\times K)\otimes\operatorname{Vect}_{\chi}\simeq D(K\times
K)\xrightarrow{\operatorname{mult}_{*}}D(K)\xrightarrow{-\overset{!}{\otimes}\chi}D(K)\xrightarrow{\Gamma_{\operatorname{dR}}}\operatorname{Vect}_{\chi}.$

For a $D(K)$-module $\mathscr{C}$, one then has the twisted invariants
$\mathscr{C}^{K,\chi}$ and coinvariants $\mathscr{C}_{K,\chi}$. The results of
Sections 2.4.2 and 2.4.3 carry over mutatis mutandis.

The only character sheaves we need in this paper arise as follows. On
$\mathbb{G}_{a}$, the exponential D-module exp($z$), concentrated in
cohomological degree $-1$, is a character sheaf with respect to a unique
structural isomorphism. Given a homomorphism

$\psi:K\rightarrow\mathbb{G}_{a},$

we denote by the same letter $\psi$ the character sheaf
$\psi^{!}\operatorname{exp}(z).$

Finally, we need to recall the notion of a generic additive character of a
maximal unipotent group $N$ in a connected reductive group $G$. Namely, the
adjoint action of the associated Borel $B$ on $N$ induces an action of the
abstract Cartan $T$ on the group scheme of homomorphisms

(2.5) $\operatorname{Hom}(N,\mathbb{G}_{a}).$

One says that an additive character $\psi:N\rightarrow\mathbb{G}_{a}$, viewed
as a $k$-point of (2.5), is generic if it lies in the open $T$-orbit. Plainly,
if one fixes a pinning of $G$, this identifies (2.5) with a product of
$\mathbb{G}_{a}$’s indexed by the Dynkin diagram of $G$; a character is
generic if and only if its coordinates are all nonzero.

###  2.5. Lie algebra representations

Some references for this material are [FG09a], [AG15], and [Ras20].

#### 2.5.1.

Fix an affine algebraic group $K$ of finite type. We write $\mathfrak{k}$ for
the Lie algebra of $K$, and denote the category of $\mathfrak{k}$-modules by
$\mathfrak{k}\operatorname{-mod}$. Explicitly, this is the canonical dg-
enhancement of the unbounded derived category of its abelian category of
representations.

This carries a canonical structure of $D(K)$-module, which may be seen as
follows. Consider $D(K)$ as a $D(K)$-module by right convolution, and consider
the category of weakly equivariant objects $D(K)^{K,w}.$ Taking invariant
sections of the underlying quasicoherent sheaf yields an equivalence

$\Gamma^{K}:D(K)^{K,w}\simeq\mathfrak{k}\operatorname{-mod}.$

Since this is important for us, we sketch the argument for the reader’s
convenience. Note that the left hand side is compactly generated by the
algebra of differential operators, with its standard weakly equivariant
structure. This is sent via $\Gamma^{K}$ to the enveloping algebra, and
identifies their endomorphisms, which yields the assertion.

In particular, $\mathfrak{k}\operatorname{-mod}$ acquires a $D(K)$-action via
left convolution. Concretely, the associated action map is of the form

(2.6)
$D(K)\otimes\mathfrak{k}\operatorname{-mod}\rightarrow\mathfrak{k}\operatorname{-mod},\quad
M\boxtimes N\mapsto\Gamma(K,M)\underset{\mathfrak{k}}{\otimes}N,$

where the appearing copy of $\mathfrak{k}$ under the tensor product is the
$K\times K$ invariant vertical vector fields with respect to the
multiplication map $K\times K\rightarrow K$.

Moreover, by the canonical identification of weak invariants and coinvariants,
one has for any $D(K)$-module $\mathscr{C}$ the identity

(2.7)
$\operatorname{Hom}_{D(K)\operatorname{-mod}}(\mathfrak{k}\operatorname{-mod},\mathscr{C})\simeq\mathscr{C}^{K,w}.$

By restriction, for any subgroup $P\subset K$ the category
$\mathfrak{k}\operatorname{-mod}$ is a $D(P)$-module. The category of
equivariant objects $\mathfrak{k}\operatorname{-mod}^{P}$ is canonically
equivalent to the derived category of modules for the Harish-Chandra pair
$(\mathfrak{k},P)$, cf. Appendix A of [Ras21], Section 5 of [CD19], or Section
9 of [Ras20]. In particular, if $K$ is a connected reductive group and $P$ is
a Borel subgroup, we obtain a version of Category $\mathscr{O}$, or more
precisely the part supported on the weight lattice.

#### 2.5.2.

Let us turn to the infinite-dimensional case. Suppose $K$ is a group placid
ind-scheme which is formally smooth and Tate. Recall that the latter means
that $K$ admits a closed embedding $K_{\circ}\hookrightarrow K$ where
$K_{\circ}$ is an affine group subscheme of $K$ and $K/K_{\circ}$ is an ind-
scheme of ind-finite type. We refer to such a subgroup $K_{\circ}$ as a
compact open subgroup in what follows.

Under this assumption, the topological Lie algebra $\mathfrak{k}$ of $K$ is a
Tate Lie algebra. For a fixed continuous central extension

$0\rightarrow
k\cdot\mathbf{1}\rightarrow\mathfrak{k}_{\kappa}\rightarrow\mathfrak{k}\rightarrow
0,$

its renormalized category of representations is defined as follows. Consider
the abelian category $\mathfrak{k}_{\kappa}\operatorname{-mod}^{\heartsuit}$
of smooth representations of $\mathfrak{k}_{\kappa}$ on which $\mathbf{1}$
acts by the identity. Within its bounded derived category
$\mathfrak{k}_{\kappa}\operatorname{-mod}^{b}$, consider the pretriangulated
envelope $\mathscr{S}$ of all representations induced from finite dimensional
smooth representations of compact open subalgebras. By definition,
$\mathfrak{k}_{\kappa}\operatorname{-mod}$ is the ind-completion of
$\mathscr{S}$.

Recall that $K$ admits a canonical extension by the multiplicative group,
namely its Tate extension

$1\rightarrow\mathbb{G}_{m}\rightarrow\widehat{K}\rightarrow K\rightarrow 1.$

Suppose that $\kappa$ is a $k$-multiple of the associated Tate extension of
$\mathfrak{k}$. In this case, one has a monoidal category $D_{\kappa}(K)$ of
twisted D-modules, and $\mathfrak{k}_{\kappa}\operatorname{-mod}$ is naturally
a $D_{\kappa}(K)$-module. Concretely, the action is similar to (2.6), with Lie
algebra homology replaced by semi-infinite homology, cf. Section 22 of [FG06].

For any compact open subgroup $K_{\circ}\subset K$, the twisting $\kappa$ is
canonically trivial upon restriction to $K_{\circ}$, and the category of
equivariant objects

(2.8) $\mathfrak{k}_{\kappa}\operatorname{-mod}^{K_{\circ}}$

may be described as follows. Within the bounded derived category of modules
for the Harish-Chandra pair $(\mathfrak{k}_{\kappa},K_{\circ})$, consider the
pretriangulated envelope $\mathscr{S}$ of modules induced from finite
dimensional $K_{\circ}$-modules. Then (2.8) is canonically equivalent to the
ind-completion of $\mathscr{S}$. In particular, for $K$ the loop group of an
almost simple, simply connected group, and $K_{\circ}$ the Iwahori subgroup,
we obtain a version of affine Category $\mathscr{O}$ at level $\kappa$, or
more precisely the part supported on the weight lattice $\Lambda_{f}$.

#### 2.5.3.

We now discuss two pieces of nonstandard notation which will be convenient in
what follows. Let $G$ and $\widehat{\mathfrak{h}}_{\kappa}$ be as in case (ii)
of Section 2.2. Note that if $\kappa$ is integral, then
$\widehat{\mathfrak{h}}_{\kappa}\operatorname{-mod}$ carries an action of
$D(G)$, i.e. one may trivialize the twisting. We may then define the
$D(G)$-module

(2.9) $\mathfrak{g}\operatorname{-mod}:=\underset{\kappa}{\oplus}\hskip
2.84526pt\widehat{\mathfrak{h}}_{\kappa}\operatorname{-mod},$

where $\kappa$ runs over the set $\mathscr{K}$ of noncritical integral levels,
i.e. $\mathscr{K}=(\mathbb{Z}\cdot\kappa_{b})\setminus\kappa_{c}.$ Similarly,
let us write $\Lambda$ for the set of highest weights for modules in
$\mathfrak{g}\operatorname{-mod}^{B}$. Plainly, we have

$\Lambda\simeq\Lambda_{f}\times\mathscr{K},$

where the pair $(\lambda_{f},\kappa)$ corresponds to the Verma module for
$\widehat{\mathfrak{h}}_{\kappa}$ of highest weight $\lambda_{f}$. Recall that
at an integral level $\kappa$, the dot action of the affine Weyl group $W$
preserves $\Lambda_{f}$. In particular, it makes sense to speak of the dot
action of $W$ on $\Lambda$.

As a final piece of notation related to our definition of $\Lambda$, let us
write $\rho_{f}\in\Lambda_{f}$ for the half sum of the positive roots of
$\mathfrak{h}$, and $\rho$ for the element $(\rho_{f},\kappa_{c})$ of
$\Lambda_{f}\times\mathbb{Z}\cdot\kappa_{b}$. With this, we have the
involution

$\lambda\mapsto-\lambda-2\rho:\Lambda\rightarrow\Lambda,$

which plays a basic role in cohomological duality. Explicitly, this sends a
pair $(\lambda_{f},\kappa)$ to $(-\lambda_{f}-2\rho_{f},-\kappa+2\kappa_{c})$.

#### 2.5.4.

It will be important for us that the usual functors of induction and
restriction of Lie algebra representations are maps of categorical group
representations. Since we are unsure if this already appears in the
literature, we include proofs.

Suppose $K_{1},K_{2}$ are a pair of groups as in Section 2.5.2, with
corresponding Tate Lie algebras $\mathfrak{k}_{1},\mathfrak{k}_{2}$. Suppose
$h:K_{1}\rightarrow K_{2}$ is a homomorphism of group ind-schemes and
$\mathfrak{k}_{2,\kappa}$ is a central extension of $\mathfrak{k}_{2}$ such
that it and its pullback $\mathfrak{k}_{1,\kappa}$ to $\mathfrak{k}_{1}$ are
$k$-multiples of the Tate extension.

######  Lemma 2.5.5.

The associated restriction functor

$\operatorname{Res}:\mathfrak{k}_{2,\kappa}\operatorname{-mod}\rightarrow\mathfrak{k}_{1,\kappa}\operatorname{-mod}$

carries a canonical datum of $D_{\kappa}(K_{1})$-equivariance.

###### Proof.

Following the notation of Section 9 of [Ras20], restriction identifies with
the composition

$\mathfrak{k}_{2,\kappa}\operatorname{-mod}\simeq
D^{!}_{\kappa}(K_{2})^{K_{2},w}\xrightarrow{\operatorname{Oblv}}D^{!}_{\kappa}(K_{2})^{K_{1},w}\xrightarrow{h^{!}}D^{!}_{\kappa}(K_{1})^{K_{1},w}\simeq\mathfrak{k}_{1,\kappa}\operatorname{-mod},$

and the two intermediate functors carry canonical data of
$D_{\kappa}(K_{1})$-equivariance. ∎

By passing to left adjoints, we obtain the following.

######  Corollary 2.5.6.

If $K_{1}$ is a compact open subgroup of $K_{2}$, the functor of induction

$\operatorname{ind}:\mathfrak{k}\operatorname{-mod}\rightarrow\mathfrak{k}_{2,\kappa}\operatorname{-mod}$

carries a canonical datum of $D(K_{1})$-equivariance.

#### 2.5.7.

As for D-modules, the appearing categories of Lie algebra representations are
canonically self-dual. Namely, for any finite dimensional Lie algebra
$\mathfrak{k}$, the functor of Lie algebra homology

$\mathfrak{k}\operatorname{-mod}\otimes\hskip
1.42262pt\mathfrak{k}\operatorname{-mod}\rightarrow\operatorname{Vect},\quad\quad
M\boxtimes N\mapsto M\underset{\mathfrak{k}}{\otimes}N$

is a perfect pairing. In particular, one has a canonical equivalence of
monoidal $\infty$-categories

$\mathfrak{k}\oplus\mathfrak{k}\operatorname{-mod}\simeq\mathfrak{k}\operatorname{-mod}\otimes\hskip
1.42262pt\mathfrak{k}\operatorname{-mod}\simeq\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\mathfrak{k}\operatorname{-mod},\mathfrak{k}\operatorname{-mod}),$

cf. (2.1). Plainly, this exchanges the tensor product of bimodules with the
composition of endofunctors.

Let us turn to the affine Lie algebra $\widehat{\mathfrak{h}}_{\kappa}$, cf.
Section 2.2. Consider the functor of semi-infinite homology, which we
normalize with respect to the pro-Lie algebra of its Iwahori subgroup,

$\widehat{\mathfrak{h}}_{\kappa}\operatorname{-mod}\otimes\hskip
1.42262pt\widehat{\mathfrak{h}}_{-\kappa+2\kappa_{c}}\operatorname{-mod}\rightarrow\operatorname{Vect},\quad
M\boxtimes N\mapsto
M\underset{\widehat{\mathfrak{h}}}{\overset{\frac{\infty}{2}}{\otimes}}\hskip
1.42262ptN.$

It was shown by Arkhipov–Gaitsgory that this is a perfect pairing [AG15]; for
an alternative proof cf. Section 5 of [Dhi21]. By Section 9 of [Ras20] this
functor carries a canonical datum of $D_{\kappa}(G)$-equivariance, i.e. the
duality is one of categorical representations.

######  Remark 2.5.8.

We remind that the Lie algebra of the Iwahori only appears to normalize the
functor; any other compact open subalgebra would do, and the resulting functor
would differ by tensoring with a cohomologically graded line.

One again has a canonical equivalence of monoidal $\infty$-categories

$\widehat{\mathfrak{h}}_{\kappa}\oplus\widehat{\mathfrak{h}}_{-\kappa+2\kappa_{c}}\operatorname{-mod}\simeq\widehat{\mathfrak{h}}_{\kappa}\operatorname{-mod}\otimes\hskip
1.42262pt\widehat{\mathfrak{h}}_{-\kappa+2\kappa_{c}}\operatorname{-mod}\simeq\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\widehat{\mathfrak{h}}_{\kappa}\operatorname{-mod},\widehat{\mathfrak{h}}_{\kappa}\operatorname{-mod});$

this exchanges the semi-infinite tensoring of bimodules with the composition
of endo-functors.

By summing over the noncritical integral levels $\mathscr{K}$, and recalling
that $\kappa_{c}$ is integral, it follows that
$\mathfrak{g}\operatorname{-mod}$ is self-dual, and that one has a canonical
equivalence

$\mathfrak{g}\operatorname{-mod}\otimes\hskip
1.42262pt\mathfrak{g}\operatorname{-mod}\simeq\operatorname{Hom}_{\operatorname{DGCat}_{\operatorname{cont}}}(\mathfrak{g}\operatorname{-mod},\mathfrak{g}\operatorname{-mod}).$

Concretely, to compose bimodules

$M\in\widehat{\mathfrak{h}}_{\kappa_{1}}\oplus\widehat{\mathfrak{h}}_{\kappa_{2}}\operatorname{-mod},\quad\quad
N\in\widehat{\mathfrak{h}}_{\kappa_{3}}\oplus\widehat{\mathfrak{h}}_{\kappa_{4}}\operatorname{-mod},$

one tensors them if the anomaly vanishes, and otherwise sets their composition
to zero, i.e.

$M\star
N:=\begin{cases}M\underset{\widehat{\mathfrak{h}}}{\overset{\frac{\infty}{2}}{\otimes}}\hskip
1.42262ptN\in\widehat{\mathfrak{h}}_{\kappa_{1}}\oplus\widehat{\mathfrak{h}}_{\kappa_{4}}\operatorname{-mod}&\operatorname{if}\kappa_{2}+\kappa_{3}=2\kappa_{c},\\\
0&\operatorname{otherwise.}\end{cases}$

##  3\. Hecke algebras and invariant vectors

In this section we record, in the present categorical context, the fundamental
adjunctions between representations of a group and its Hecke algebras, as well
as their basic properties.

The statements and proofs in this section have the flavor of abstract
nonsense. The reader mostly interested in representation-theoretic
applications may wish to glance at the statements of Theorem 3.1.5, Definition
3.2.2, and Corollary 3.2.4 before proceeding directly to Section 4.

As with later sections containing abstract nonsense, many of the basic results
are known to experts.

#### 3.0.1.

We first recall the analogous theory for actions of groups on vector spaces;
the reader may wish to skip this and proceed directly to Section 3.1.

Suppose that $H$ is a finite group, and write $H\operatorname{-mod}$ for its
category of representations over a field of characteristic zero. Given a
subgroup $K$ of $H$, one can consider the associated functor of
$K$-invariants. This is co-represented by the space of functions
$\operatorname{Fun}(H/K)$, whose endomorphisms are the convolution algebra
$\operatorname{Fun}(K\backslash H/K).$ In particular, we obtain an adjunction

(3.1) $\operatorname{Fun}(H/K)\underset{\operatorname{Fun}(K\backslash
H/K)}{\otimes}-:\operatorname{Fun}(K\backslash
H/K)\operatorname{-mod}\rightleftarrows
H\operatorname{-mod}:\operatorname{inv}^{K}.$

The left adjoint in (3.1) is fully faithful, with essential image generated by
the simple representations which have nontrivial $K$ invariants. In
particular, for a representation $\pi$ of $H$, the co-unit

(3.2) $\operatorname{Fun}(H/K)\hskip
14.22636pt\underset{\operatorname{Fun}(K\backslash
H/K)}{\otimes}\pi^{K}\rightarrow\pi$

is an embedding, with essential image the corresponding isotypic components of
$\pi$, i.e. the submodule generated by $K$ invariants.

We now turn to the analogous results for categorical representations.

######  Remark 3.0.2.

For a $p$-adic group $H$ with a compact open subgroup $K$, an adjunction
similar to (3.1) plays a fundamental role in its representation theory. While
this is a more direct analogue of our study of affine Hecke categories and
loop groups, the categorical theory behaves more similarly to the case of a
finite group, including crucially the fully faithfulness of (3.2).

###  3.1. The basic adjunctions

#### 3.1.1.

Suppose $K\rightarrow H$ is a homomorphism of group placid ind-schemes, cf.
Section 2.3.1. For $D(H)$-modules, the functor of $K$ invariants is given by

$\operatorname{Hom}_{D(K)\operatorname{-mod}}(\operatorname{Vect},-)\simeq\operatorname{Hom}_{D(H)\operatorname{-mod}}(D(H)_{K},-).$

In particular, such invariants carry a right action by

$\mathscr{H}:=\operatorname{Hom}_{D(H)\operatorname{-mod}}(D(H)_{K},D(H)_{K}),$

and we obtain a tensor-hom adjunction

(3.3) $-\underset{\mathscr{H}}{\otimes}\hskip
1.42262ptD(H)_{K}:\operatorname{mod-}\mathscr{H}\rightleftarrows
D(H)\operatorname{-mod}:-^{K}.$

In this generality, the right adjoint in (3.3) need not be
$\operatorname{DGCat}_{\operatorname{cont}}$-linear, nor commute with
colimits.

#### 3.1.2.

Let us further suppose that $K$ is a affine group subscheme of $H$ whose
prounipotent radical is of finite codimension. In this case, one has a
canonical identification of invariants and coinvariants with respect to $K$,
cf. Section 2.4.3. In particular, we may canonically identify $\mathscr{H}$
with the convolution algebra of bi-$K$-equivariant D-modules

$\mathscr{H}\simeq D(K\backslash H/K).$

Inversion on $H$ induces a canonical isomorphism between $\mathscr{H}$ and its
opposite algebra, and in particular between its categories of left and right
modules. We use both of these identifications implicitly going forwards.

######  Proposition 3.1.3.

Under our assumption on $K$, the left adjoint in (3.3) is fully faithful.

###### Proof.

By its identification with $K$ coinvariants, the functor of $K$ invariants
commutes with colimits and is
$\operatorname{DGCat}_{\operatorname{cont}}$-linear. It follows, e.g. via
using the bar resolution, that for any $\mathscr{H}$-module $\mathscr{C}$ the
unit identifies with the composition

$\mathscr{C}\simeq D(K\backslash H/K)\underset{D(K\backslash
H/K)}{\otimes}\mathscr{C}\simeq\big{(}D(H/K)\underset{D(K\backslash
H/K)}{\otimes}\mathscr{C}\big{)}^{K}.$

∎

#### 3.1.4.

We continue to assume that $K$ is as in Section 3.1.2, and make the further
assumption that $H/K$ is ind-proper (and in particular, of ind-finite type).
This yields the following property of the counit, which will be important in
what follows.

######  Theorem 3.1.5.

With our assumptions on $K$, for any $D(H)$-module $\mathscr{C}$, the counit

(3.4) $D(H/K)\underset{D(K\backslash
H/K)}{\otimes}\mathscr{C}^{K}\xrightarrow{\hskip 5.69054ptc\hskip
5.69054pt}\mathscr{C}$

is fully faithful and admits a (continuous) $D(H)$-equivariant right adjoint
$c^{R}$.

Informally, the theorem allows one to reconstruct a piece of the entire module
from its $K$ invariants, which are often easier to analyze directly. The
result and its proof are similar to work of Ben-Zvi–Gunningham–Orem [BZGO18];
for a generalization see also the interesting paper of Yang [Yan21].

Before giving the proof, we mention an important special case.

######  Example 3.1.6.

Suppose that $K=H$. In this case we refer to the obtained category as the
monodromic invariants, i.e.

(3.5) $\displaystyle\mathscr{C}^{K\operatorname{-mon}}$
$\displaystyle:=\mathscr{C}^{K}\underset{D(K\backslash K/K)}{\otimes}D(K/K)$
(3.6)
$\displaystyle\simeq\mathscr{C}^{K}\underset{D(\operatorname{pt}/K)}{\otimes}\operatorname{Vect}.$

Theorem 3.1.5 identifies this with the full subcategory of $\mathscr{C}$
generated under colimits by the essential image of the forgetful functor
$\operatorname{Oblv}:\mathscr{C}^{K}\rightarrow\mathscr{C}.$

So, in this case Theorem 3.1.5 reconstructs a ‘formal neighborhood’ of
$\mathscr{C}^{K}$ inside of $\mathscr{C}$ using the action of
$D(\operatorname{pt}/K)$ on the former. This may be thought of as a version of
de-equivariantization.

###### Proof of Theorem 3.1.5.

The claimed equivariance of such a right adjoint is automatic for
$D(H)$-modules, cf. Section 2.4.1.

By the $\operatorname{DGCat}_{\operatorname{cont}}$-linearity and continuity
of the appearing functors, it suffices to show that the $D(H)\otimes
D(H)$-equivariant functor

(3.7) $D(H/K)\underset{D(K\backslash H/K)}{\otimes}D(K\backslash H)\rightarrow
D(H)$

is fully faithful and admits a right adjoint. Namely, the general case follows
by tensoring over $D(H)$ with $\mathscr{C}$.

We will prove the assertions for (3.7) by an analysis of the situation on
zero-simplices in the bar resolution, along with a monadicity argument.
Consider the convolution correspondence

${H\overset{K}{\times}H}$${H/K\times K\backslash
H}$${H.}$$\scriptstyle{p}$$\scriptstyle{q}$

Since $p$ is a $K$-torsor, the functor $p_{*}$ admits a left adjoint $p^{*}$
(see [Ras15], Proposition 6.18.1). With this, by definition the $D(H)\otimes
D(H)$-equivariant composition

$D(H/K)\otimes D(K\backslash H)\rightarrow D(H/K)\underset{D(K\backslash
H/K)}{\otimes}D(K\backslash H)\rightarrow D(H)$

identifies equivariantly with the composition

(3.8) $m:D(H/K)\otimes D(K\backslash H)\stackrel{{\scriptstyle
p^{*}}}{{\longrightarrow}}D(H\overset{K}{\times}H)\stackrel{{\scriptstyle
q_{*}}}{{\longrightarrow}}D(H).$

Since $q$ is ind-proper of ind-finite type, the $D(H)\otimes D(H)$-equivariant
functor $q_{*}$ admits a right adjoint $q^{!}$. By composition, it follows
that $m$ admits a right adjoint $m^{R}$.

Let us now consider the situation after tensoring over $D(K\backslash H/K)$.
We first claim that the insertion

$\operatorname{ins}:D(H/K)\otimes D(K\backslash H)\rightarrow
D(H/K)\underset{D(K\backslash H/K)}{\otimes}D(K\backslash H)$

is monadic. More precisely, we claim it admits a conservative right adjoint
$\operatorname{ins}^{R}$.

To see this, consider the presentation of $D(H/K)\underset{D(K\backslash
H/K)}{\otimes}D(K\backslash H)$ via the semisimplicial bar resolution

(3.9) ${\cdots}$${D(H/K)\otimes D(K\backslash H/K)\otimes D(K\backslash
H)}$${D(H/K)\otimes D(K\backslash H).}$

It follows from the preceding paragraph that each face map admits a right
adjoint. Under the canonical equivalence of the colimit with the limit over
its right adjoints, $\operatorname{ins}^{R}$ identifies with evaluation on
zero-simplices, which yields the desired claim.

In term of the bar resolution, (3.7) is given by the augmented semisimplicial
dg-category

${\cdots D(H/K)\otimes D(K\backslash H/K)\otimes D(K\backslash
H)}$${D(H/K)\otimes D(K\backslash H)}$${D(H).}$$\scriptstyle{q_{*}p^{*}}$

Thus the functor (3.7) amounts to a morphism of monads

$\operatorname{ins}^{R}\operatorname{ins}\longrightarrow
p_{*}q^{!}q_{*}p^{*}.$

We will prove that this is an isomorphism, which is equivalent to the fully
faithfulness of (3.7).

Corollary C.2.3 in [Gai15] says that the monad
$\operatorname{ins}^{R}\operatorname{ins}$ identifies with

$\displaystyle D(H/K)\otimes D(K\backslash H)$
$\displaystyle\xrightarrow{\text{id}_{D(H/K)}\otimes\operatorname{act}_{D(K\backslash
H)}^{R}}D(H/K)\otimes D(K\backslash H/K)\otimes D(K\backslash H)$
$\displaystyle\xrightarrow{\operatorname{act}_{D(H/K)}\otimes\text{id}_{D(K\backslash
H)}}D(H/K)\otimes D(K\backslash H)$

as an endofunctor. Thus it is given by pull-push along the correspondence

${H\overset{K}{\times}H\overset{K}{\times}H}$${H/K\times K\backslash
H\overset{K}{\times}H}$${H\overset{K}{\times}H/K\times K\backslash
H}$${H/K\times K\backslash H}$${H/K\times K\backslash H/K\times K\backslash
H}$${H/K\times K\backslash H.}$

But the endofunctor $p_{*}q^{!}q_{*}p^{*}$ is given by pull-push along the
isomorphic correspondence

${H\overset{K}{\times}H\overset{K}{\times}H}$${H\overset{K}{\times}H}$${H\overset{K}{\times}H}$${H/K\times
K\backslash H}$${H}$${H/K\times K\backslash H,}$

and a diagram chase shows that the resulting natural isomorphism
$\operatorname{ins}^{R}\operatorname{ins}\tilde{\to}p_{*}q^{!}q_{*}p^{*}$
identifies with the aforementioned morphism of monads. This completes the
proof of the fully faithfulness of (3.7).

It remains to furnish (3.7) with a right adjoint $\eqref{e:univcase}^{R}$. To
see this, recall that the morphism $m$ identifies with the composition

$D(H/K)\otimes D(K\backslash
H)\xrightarrow{\operatorname{ins}}D(H/K)\underset{D(K\backslash
H/K)}{\otimes}D(K\backslash H)\xrightarrow{\eqref{e:univcase}}D(H).$

The desired continuity of $\eqref{e:univcase}^{R}$, which exists as a
discontinuous functor by the adjoint functor theorem, follows via the
conservativity and continuity of $\operatorname{ins}^{R}$ from the continuity
of $m^{R}$. ∎

#### 3.1.7.

Let us deduce a useful consequence of Theorem 3.1.5. In particular, we keep
its assumptions on $K$.

######  Corollary 3.1.8.

For $K$ as in Theorem 3.1.5, the functor $D(H/K)\underset{D(K\backslash
H/K)}{\otimes}-$ is also right adjoint to the functor of $K$ invariants, with
unit

$\mathscr{C}\xrightarrow{\hskip 2.84526ptc^{R}\hskip
2.84526pt}D(H/K)\underset{D(K\backslash H/K)}{\otimes}\mathscr{C}^{K}.$

###### Proof.

For a $D(H)$-module $\mathscr{C}$ and $D(K\backslash H/K)$-module
$\mathscr{M}$, we have a map

$\displaystyle\operatorname{Hom}_{D(K\backslash
H/K)\operatorname{-mod}}(\mathscr{C}^{K},\mathscr{M})$
$\displaystyle\overset{\ref{p:fftensoringup}}{\simeq}\operatorname{Hom}_{D(H)\operatorname{-mod}}\big{(}D(H/K)\underset{D(K\backslash
H/K)}{\otimes}\mathscr{C}^{K},D(H/K)\underset{D(K\backslash
H/K)}{\otimes}\mathscr{M}\big{)}$ $\displaystyle\hskip
2.84526pt\rightarrow\operatorname{Hom}_{D(H)\operatorname{-mod}}\big{(}\mathscr{C},D(H/K)\underset{D(K\backslash
H/K)}{\otimes}\mathscr{M}\big{)}.$

By Theorem 3.1.5, the latter arrow is a full embedding. Its essential
surjectivity follows from the naturality of $c^{R}$, as it is an equivalence
when applied to $D(H/K)\underset{D(K\backslash H/K)}{\otimes}\mathscr{M}$. ∎

######  Remark 3.1.9.

Note that Corollary 3.1.8 can be rephrased as a natural isomorphism of
functors

(3.10) $D(H/K)\underset{D(K\backslash H/K)}{\otimes}(-)\hskip
5.69054pt\simeq\hskip 5.69054pt\operatorname{Hom}_{D(K\backslash
H/K)\operatorname{-mod}}(D(H/K),-).$

###  3.2. Generation

#### 3.2.1.

Let $H$ and $K$ be as in Theorem 3.1.5. In the remainder of this section, we
collect some tools which allow one to control a categorical representation
from its $K$ invariants. The basic definition is the following.

######  Definition 3.2.2.

For a $D(H)$-module $\mathscr{C}$, we write
$\mathscr{C}^{K\operatorname{-gen}}$ for the submodule of $\mathscr{C}$
generated by its $K$ invariants, i.e. the essential image of the counit (3.4):

$D(H/K)\underset{D(K\backslash H/K)}{\otimes}\mathscr{C}^{K}\hskip
2.84526pt\overset{\sim}{\longrightarrow}\hskip
2.84526pt\mathscr{C}^{K\operatorname{-gen}}\hskip 2.84526pt\subset\hskip
2.84526pt\mathscr{C}.$

We say that $\mathscr{C}$ is generated by its $K$ invariants if
$\mathscr{C}^{K\operatorname{-gen}}=\mathscr{C}$, i.e. if the inclusion is an
equivalence.

#### 3.2.3.

Suppose ${\Theta}:\mathscr{C}\rightarrow\mathscr{D}$ is a morphism of
$D(H)$-modules, and consider the associated morphism of $K$-invariant
categories

${\Theta}^{K}:\mathscr{C}^{K}\rightarrow\mathscr{D}^{K}.$

If ${\Theta}$ is fully faithful, it formally follows that ${\Theta}^{K}$ is as
well. Indeed, given a general monoidal category $\mathscr{M}$, a fully
faithful map of $\mathscr{M}$-modules $\mathscr{C}\rightarrow\mathscr{D}$, and
another $\mathscr{M}$-module $\mathscr{S}$, the map

(3.11)
$\operatorname{Hom}_{\mathscr{M}\operatorname{-mod}}(\mathscr{S},\mathscr{C})\rightarrow\operatorname{Hom}_{\mathscr{M}\operatorname{-mod}}(\mathscr{S},\mathscr{D})$

is a fully faithful embedding.

The following partial converse, which we will make significant use of in what
follows, allows one to check fully faithfulness, and in particular
equivalences, on ‘highest weight vectors’.

######  Corollary 3.2.4.

If $\mathscr{C}$ is generated by $K$ invariants and ${\Theta}^{K}$ is fully
faithful, then ${\Theta}$ also is fully faithful.

###### Proof.

By our assumption on $\mathscr{C}$, we obtain a canonical factorization of
$\Theta$ as a composition

(3.12)
$\mathscr{C}^{K\operatorname{-gen}}\rightarrow\mathscr{D}^{K\operatorname{-gen}}\rightarrow\mathscr{D}.$

By Theorem 3.1.5 the second arrow of (3.12) is fully faithful, so it is enough
to show the first arrow of (3.12) is also fully faithful. However, as in
Remark 3.1.9 we may rewrite this as

$\operatorname{Hom}_{D(K\backslash
H/K)\operatorname{-mod}}\big{(}D(H/K),\mathscr{C}^{K}\big{)}\rightarrow\operatorname{Hom}_{D(K\backslash
H/K)\operatorname{-mod}}\big{(}D(H/K),\mathscr{D}^{K}\big{)},$

hence its fully faithfulness follows from (3.11). ∎

Let us mention two other consequences of Theorem 3.1.5, namely the favorable
interaction of generation by $K$ invariants with passing to subcategories and
gluing. Since we do not use these in what follows, we omit their formal
deductions.

######  Corollary 3.2.5.

If $\mathscr{C}$ is generated by $K$ invariants, and
$\mathring{\mathscr{C}}\rightarrow\mathscr{C}$ is a fully faithful embedding
of $D(H)$-modules, then $\mathring{\mathscr{C}}$ is generated by $K$
invariants.

######  Corollary 3.2.6.

Suppose $\mathscr{M}^{\prime}$, $\mathscr{M}$ and $\mathscr{M}^{\prime\prime}$
are $D(H)$-modules fitting into a recollement

$\mathscr{M}^{\prime}\rightleftarrows\mathscr{M}\rightleftarrows\mathscr{M}^{\prime\prime}.$

Then $\mathscr{M}$ is generated by $K$ invariants if and only if
$\mathscr{M}^{\prime}$ and $\mathscr{M}^{\prime\prime}$ are.

#### 3.2.7.

Finally, let us note the compatibility of $K$ generation and duality. Recall
that if $\mathscr{C}$ is a $D(H)$-module which is dualizable as an object of
$\operatorname{DGCat}_{\operatorname{cont}}$, its dual $\mathscr{C}^{\vee}$
carries a canonical structure of $D(H)$-module.

######  Corollary 3.2.8.

For $H$ and $K$ as in Theorem 3.1.5, and a dualizable $D(H)$-module
$\mathscr{C}$, there are canonical isomorphisms

$(\mathscr{C}^{K\operatorname{-gen}})^{\vee}\simeq(\mathscr{C}^{\vee})^{K\operatorname{-gen}}.$

###### Proof.

Recall the adjunction of Theorem 3.1.5, i.e.
$\mathscr{C}^{K\operatorname{-gen}}\rightleftarrows\mathscr{C}.$ The fully
faithfulness of the left adjoint and the dualizability of $\mathscr{C}$
straightforwardly imply the dualizability of
$\mathscr{C}^{K\operatorname{-gen}}$. By our assumption on $K$, duality
interchanges the adjunctions

(3.13)
$\operatorname{Oblv}:\mathscr{C}^{K}\rightleftarrows\mathscr{C}:\operatorname{Av}^{H}_{*}\quad\quad\text{and}\quad\quad\operatorname{Oblv}:(\mathscr{C}^{\vee})^{K}\rightleftarrows\mathscr{C}^{\vee}:\operatorname{Av}^{H}_{*}.$

It follows that, viewed as full subcategories of $\mathscr{C}^{\vee}$, we have

(3.14)
$(\mathscr{C}^{\vee})^{K\operatorname{-gen}}\subset(\mathscr{C}^{K\operatorname{-gen}})^{\vee}.$

Interchanging the roles of $\mathscr{C}$ and $\mathscr{C}^{\vee}$, we
similarly have

(3.15)
$\mathscr{C}^{K\operatorname{-gen}}\subset((\mathscr{C}^{\vee})^{K\operatorname{-gen}})^{\vee}.$

Comparing (3.14) and (3.15), it follows both inclusions are equivalences, as
desired. ∎

##  4\. A category of Lie algebra representations

In this section, we record the definition and first properties of the
categories of Lie algebra representations considered in this paper.

#### 4.0.1.

Recall that $G$ denotes a reductive group or loop group, $B$ denotes a Borel
subgroup of $G$, $\mathfrak{g}$ denotes the Lie algebra of $G$, and
$\mathfrak{g}\operatorname{-mod}$ denotes its category of representations, cf.
Sections 2.2 and 2.5 for our precise conventions.

#### 4.0.2.

Recall that $\mathfrak{g}\operatorname{-mod}$ carries a canonical action of
$D(G)$, cf. Section 2.5. So, as in Definition 3.2.2 we may consider

$\mathfrak{g}\operatorname{-mod}^{B\operatorname{-gen}}:=D(G/B)\underset{D(B\backslash
G/B)}{\otimes}\mathfrak{g}\operatorname{-mod}^{B}\hskip 5.69046pt\subset\hskip
5.69046pt\mathfrak{g}\operatorname{-mod}.$

We next isolate certain submodules. To do so, we begin with some preliminary
generalities.

#### 4.0.3.

Given a $D(G)$-module $\mathscr{C}$ and object $\xi$ of $\mathscr{C}$, one may
consider the $D(G)$-submodule generated by $\xi$, i.e. the minimal full
subcategory of $\mathscr{C}$ stable under the action of $D(G)$ which contains
$c$. Explicitly, convolution with $\xi$ gives an equivariant functor

(4.1) $D(G)\rightarrow\mathscr{C},$

and the desired full subcategory of $\mathscr{C}$ is the closure of the
essential image of (4.1) under colimits.

######  Lemma 4.0.4.

Suppose that $K$ is a subgroup of $G$ for which $G/K$ is ind-proper, and that
$\xi$ is equipped with a datum of $K$-equivariance. If we write $\mathscr{S}$
for the $D(K\backslash G/K)$-submodule of $\mathscr{C}^{K}$ generated by
$\xi$, then the $D(G)$-submodule generated by $\xi$ is canonically equivalent
to

(4.2) $D(G/K)\underset{D(K\backslash G/K)}{\otimes}\mathscr{S}.$

###### Proof.

This follows from Theorem 3.1.5. ∎

#### 4.0.5.

Recall the set of integral weights $\Lambda$, and that
$\mathfrak{g}\operatorname{-mod}^{B}$ is compactly generated by the Verma
modules $M_{\lambda}$, for $\lambda\in\Lambda$, cf. Sections 2.2 and 2.5.

######  Definition 4.0.6.

For $\lambda\in\Lambda$, denote the $D(G)$-submodule of
$\mathfrak{g}\operatorname{-mod}$ generated by $M_{\lambda}$ by

$i_{!}:\mathfrak{g}\operatorname{-mod}_{\lambda}\rightarrow\mathfrak{g}\operatorname{-mod}.$

We next identify compact generators for the $B$ invariants of
$\mathfrak{g}\operatorname{-mod}_{\lambda}$.

######  Proposition 4.0.7.

The category $\mathfrak{g}\operatorname{-mod}_{\lambda}^{B}$ is compactly
generated by the Verma modules $M_{w\cdot\lambda}$ for $w\in W$. In
particular, the category $\mathfrak{g}\operatorname{-mod}_{\lambda}$ only
depends on the Weyl group dot orbit of $\lambda$.

###### Proof.

For a simple reflection $s$ of $W$, write $j_{s,!}$ for the corresponding
standard object and $j_{s,*}$ for the corresponding costandard object of
$D(B\backslash G/B)$. For a $D(G)$-module $\mathscr{C}$, let us denote the
underlying binary product of the action of $D(B\backslash G/B)$ on
$\mathscr{C}^{B}$ by

(4.3) $-\star-:D(B\backslash
G/B)\otimes\mathscr{C}^{B}\rightarrow\mathscr{C}^{B},\quad\quad M\boxtimes
c\mapsto M\star c.$

We will deduce the proposition from the following lemma.

######  Lemma 4.0.8.

Fix a simple reflection $s$ and a weight $\nu\in\Lambda$. We have

(4.4) $\displaystyle j_{s,!}\star M_{\nu}\simeq
M_{s\cdot\nu}\quad\quad\quad\text{ if
$\langle\check{\alpha}_{s},\nu+\rho\rangle\leqslant 0.$}$ (4.5) $\displaystyle
j_{s,*}\star M_{\nu}\simeq M_{s\cdot\nu}\quad\quad\quad\text{if
$\langle\check{\alpha}_{s},\nu+\rho\rangle\geqslant 0$}.$

######  Remark 4.0.9.

The result of the lemma is well known, albeit typically phrased in the
language of twisting functors. However, the fact that it tautologically
follows from the case of $\mathfrak{sl}_{2}$ via categorical equivariance
properties, as we show below, appears to be new.

###### Proof.

Write $P_{s}$ for the parabolic associated to $s$, and $G_{s}$ for its Levi
quotient, which is of semisimple rank one. Writing $B_{s}$ for its induced
Borel subgroup, by the $D(B_{s}\backslash G_{s}/B_{s})$-equivariance of

$\operatorname{pind}:\mathfrak{g}_{s}\operatorname{-mod}^{B_{s}}\rightarrow\mathfrak{g}\operatorname{-mod}^{B},$

it suffices to establish the lemma for $G_{s}$. In particular, for the
remainder of the lemma when we refer to the Verma modules $M_{\nu}$ and their
simple quotients, we are referring to those for $G_{s}$.

As $j_{s,!}$ and $j_{s,*}$ are inverse functors, it is straightforward to see
that (4.4) and (4.5) are equivalent. To prove (4.4), recall the triangle

(4.6) $\underline{k}\rightarrow j_{e,*}\rightarrow j_{s,!}\xrightarrow{+1},$

where $\underline{k}$ denotes the constant sheaf on $B_{s}\backslash
G_{s}/B_{s}$.For any $D(G_{s})$-module $\mathscr{C}$, one has an adjunction

$\operatorname{Oblv}:\mathscr{C}^{G_{s}}\rightleftarrows\mathscr{C}^{B_{s}}:\operatorname{Av}_{*}$

and it straightforward to see that the counit
$\operatorname{Oblv}\operatorname{Av}_{*}\rightarrow\operatorname{id}$
identifies with convolution by $\underline{k}\rightarrow j_{e,*}$ above.

Let us prove (4.4). In the singular case, i.e. if
$\langle\check{\alpha}_{s},\nu+\rho\rangle=0$, we are done from (4.6) as
$\operatorname{Av}_{*}M_{\nu}$ vanishes. If
$\langle\check{\alpha}_{s},\nu+\rho\rangle<0$, the desired conclusion follows
from rotating (4.6) to obtain a triangle

(4.7) $M_{\nu}\rightarrow j_{s,!}\star M_{\nu}\rightarrow
L_{s\cdot\nu}\otimes\operatorname{Hom}_{\mathfrak{g}_{s}\operatorname{-mod}^{B_{s}}}(L_{s\cdot\nu},M_{\nu})[1]\xrightarrow{+1},$

and noting that $j_{s,!}\star M_{\nu}$ must be indecomposable by the
invertibility of $j_{s,!}$. ∎

Write $\mathscr{S}$ for the subcategory of
$\mathfrak{g}\operatorname{-mod}^{B}$ generated the Verma modules
$M_{w\cdot\lambda}$, for $w\in W$. Using Lemma 4.0.8, it is straightforward to
see that $\mathscr{S}$ is preserved by the action of $D(B\backslash G/B)$, and
moreover is generated by $M_{\lambda}$ as a $D(B\backslash G/B)$-module. We
are therefore done by Lemma 4.0.4. ∎

######  Corollary 4.0.10.

The inclusion of $\mathfrak{g}\operatorname{-mod}_{\lambda}$ admits a right
adjoint $i^{!}$.

(4.8)
$i_{!}:\mathfrak{g}\operatorname{-mod}_{\nu}\rightleftarrows\mathfrak{g}\operatorname{-mod}:i^{!}.$

######  Remark 4.0.11.

We remind that $i^{!}$, as with any adjoint of a $D(G)$-equivariant functor,
inherits a datum of strict $D(G)$-equivariance, cf. Section 2.4.1.

###### Proof.

By Theorem 3.1.5, the inclusion
$\mathfrak{g}\operatorname{-mod}^{B\operatorname{-gen}}\rightarrow\mathfrak{g}\operatorname{-mod}$
admits a right adjoint, so it is enough to furnish a right adjoint for

$\mathfrak{g}\operatorname{-mod}_{\lambda}\rightarrow\mathfrak{g}\operatorname{-mod}^{B\operatorname{-gen}}.$

Equivalently, by Proposition 3.1.3, we must produce a $D(B\backslash
G/B)$-equivariant right adjoint to
$\mathfrak{g}\operatorname{-mod}_{\lambda}^{B}\rightarrow\mathfrak{g}\operatorname{-mod}^{B}$.
However, from Proposition 4.0.7 it is clear the inclusion preserves compact
objects, and hence admits a continuous right adjoint $i^{!}$. Its equivariance
follows from the semi-rigidity of $D(B\backslash G/B)$, cf. Lemma 3.5 of
[BZN09]. ∎

#### 4.0.12.

Let us establish some notation for later use. For a $D(G)$-module
$\mathscr{C}$, let us define

$\mathscr{C}^{G,w,\nu}:=\operatorname{Hom}_{D(G)\operatorname{-mod}}(\mathfrak{g}\operatorname{-mod}_{\nu},\mathscr{C}).$

By concatenating (2.7) and (4.8), we deduce an adjunction with fully faithful
left adjoint

(4.9) $i_{!}:\mathscr{C}^{G,w,\nu}\rightleftarrows\mathscr{C}^{G,w}:i^{!}.$

#### 4.0.13.

Recall that $\mathfrak{g}\operatorname{-mod}$ is self-dual as an abstract
category, cf. Section 2.5 for our normalizations. As a final property, let us
describe how the above categories behave under this duality. To state this,
recall that given an adjunction between dualizable categories

$F:\mathscr{C}\rightleftarrows\mathscr{D}:G,$

one canonically obtains a dual adjunction between the dual functors

$G^{\vee}:\mathscr{C}^{\vee}\rightleftarrows\mathscr{D}^{\vee}:F^{\vee}.$

######  Proposition 4.0.14.

For any $\lambda$, duality interchanges the adjunctions

$i_{!}:\mathfrak{g}\operatorname{-mod}_{\lambda}\rightleftarrows\mathfrak{g}\operatorname{-mod}:i^{!}\quad\text{and}\quad
i_{!}:\mathfrak{g}\operatorname{-mod}_{-\lambda-2\rho}\rightleftarrows\mathfrak{g}\operatorname{-mod}:i^{!}.$

###### Proof.

The dualizability of $\mathfrak{g}\operatorname{-mod}_{\lambda}$ follows from
Corollary 4.0.10. For a dualizable category $\mathscr{C}$, let us write
$\mathscr{C}^{c}$ for its full (non-cocomplete) subcategory of compact
objects, and let us denote by

$\mathbf{D}:\mathscr{C}^{c}\simeq(\mathscr{C}^{\vee})^{c,op}$

the associated identification of compact objects.

Recall that by Shapiro’s lemma the dual of a Verma module is again a
(cohomologically shifted) Verma module, namely

$\mathbf{D}M_{\lambda}\simeq M_{-\lambda-2\rho}\otimes\ell,$

where $\ell$ is a cohomologically graded line, see for example Lemma 9.8 of
[Dhi21]. In particular, it follows that

$\mathfrak{g}\operatorname{-mod}_{\lambda}\subset\mathfrak{g}\operatorname{-mod}_{-\lambda-2\rho}^{\vee}\quad\text{and}\quad\mathfrak{g}\operatorname{-mod}_{-\lambda-2\rho}\subset\mathfrak{g}\operatorname{-mod}_{\lambda}^{\vee}.$

By dualizing each of the two inclusions and comparing with the other, we
deduce both are equalities. ∎

##  5\. Localization I: The sign representation and the Steinberg module

In this section, we prove a localization theorem which embeds a maximally
singular category of Lie algebra representations into the category of
nondegenerate Whittaker sheaves.

#### 5.0.1.

The structure is as follows. After introducing notation, we construct a
functor of global sections via the universal properties of the appearing
$D(G)$-modules. To see it provides the desired fully faithful embedding, by
the general results of Section 3 it is enough to do so on the categories of
Borel equivariant objects.

In the present situation, the latter categories are copies of
$\operatorname{Vect}$. That is, said informally, both modules have a ‘unique
highest weight vector, up to scaling’. So, we simply need to check a certain
endofunctor of $\operatorname{Vect}$ is an equivalence, which is done in
Theorem 5.2.4. Finally, we note the alternative, tautological localization as
weakly equivariant D-modules.

###  5.1. Setup

#### 5.1.1.

In this subsection, we introduce the two sides of the localization theorem.
Let $G$ be a connected reductive group with a Borel subgroup $B$. Recall that
$\Lambda$ denotes the character lattice of the abstract Cartan, and a fix a
maximally singular weight $\lambda\in\Lambda$, i.e. a character $\lambda$ for
which $W\cdot\lambda=\lambda$. Consider the associated category
$\mathfrak{g}\operatorname{-mod}_{\lambda}$ as in Section 4.0.5. Passing to
its $B$ invariants, by Proposition 4.0.7 we have

(5.1)
$\mathfrak{g}\operatorname{-mod}_{\lambda}^{B}\simeq\operatorname{Vect},$

with distinguished generator the Verma module $M_{\lambda}$.

#### 5.1.2.

Fix a Borel $B^{-}$ in general position with respect to $B$, and write $N^{-}$
for its unipotent radical.777This passage to an opposite is inessential, and
is made to simplify notation in the proof. Let $\psi$ be a generic additive
character of $N^{-}$, cf. Section 2.4.5, and consider the associated category
of nondegenerate Whittaker D-modules

(5.2) $D(G/N^{-},\psi):=D(G)^{N^{-},\psi}.$

This is canonically a $D(G)$-module via left convolution.

Note that, if we write $\delta_{e}$ for the delta D-module at the identity of
$G$, (5.2) is generated as a $D(G)$-module by
$\delta_{\psi}:=\operatorname{Av}^{N^{-},\psi}_{*}\delta_{e}.$ Passing to $B$
invariants, it is standard that

(5.3) $D(G/N^{-},\psi)^{B}\simeq\operatorname{Vect},$

with canonical generator $\operatorname{Av}_{*}^{B}\delta_{\psi}.$

###  5.2. The localization theorem

#### 5.2.1.

We first construct a $D(G)$-equivariant functor in the direction
$D(G/N^{-},\psi)\rightarrow\mathfrak{g}\operatorname{-mod}_{\lambda}.$ To do
so, note that, by the identification of $(N^{-},\psi)$ invariants and
coinvariants, and tensor-hom adjunction, we have for any $D(G)$-module
$\mathscr{C}$ the equivalence

(5.4)
$\operatorname{Hom}_{D(G)\operatorname{-mod}}\big{(}D(G/N^{-},\psi),\hskip
1.42262pt\mathscr{C}\big{)}\simeq\mathscr{C}^{N^{-},-\psi}.$

Explicitly, this equivalence is given, from left to right, by evaluation on
$\delta_{\psi}$, cf. Section 5.1.2. Equivalently, from right to left, it is
given by convolution with the specified Whittaker equivariant object.

#### 5.2.2.

Let us write $\mathfrak{n}^{-}$ for the Lie algebra of $N^{-}$, and
$k_{-\psi}$ for its character given by the differential of $-\psi$. Consider
the universal Whittaker module

$\operatorname{ind}_{\mathfrak{n}^{-}}^{\mathfrak{g}}k_{-\psi}\in\mathfrak{g}\operatorname{-mod}^{N^{-},\psi}.$

By (5.4), it gives rise to an equivariant functor
$\mathring{\Gamma}:D(G/N^{-},\psi)\rightarrow\mathfrak{g}\operatorname{-mod}.$
It will be convenient in what follows to work instead with a cohomological
shift of $\mathring{\Gamma}$, namely

$\Gamma:=\mathring{\Gamma}\otimes\det\big{(}(\mathfrak{g}/\mathfrak{b})^{*}[-1]\big{)}.$

#### 5.2.3.

Recall that the inclusion
$\mathfrak{g}\operatorname{-mod}_{\lambda}\rightarrow\mathfrak{g}\operatorname{-mod}$
admits an equivariant right adjoint $i^{!}$, cf. Corollary 4.0.10. Consider
the composition

(5.5)
$i^{!}\circ\Gamma:D(G/N^{\text{--}},\psi)\rightarrow\mathfrak{g}\operatorname{-mod}\rightarrow\mathfrak{g}\operatorname{-mod}_{\lambda}.$

As the target is generated by $B$ invariants, by Corollary 3.1.8 it factors
through a map

(5.6)
$D(G/N^{-},\psi)^{B\operatorname{-gen}}\rightarrow\mathfrak{g}\operatorname{-mod}_{\lambda},$

which we presently show is the sought-for functor.

######  Theorem 5.2.4.

The functor (5.6) is an equivalence.

###### Proof.

By Corollary 3.2.4, it is enough to show the induced map

$\operatorname{Vect}\overset{\eqref{e:whitline}}{\simeq}D(B\backslash
G/N^{-},\psi)\rightarrow\mathfrak{g}\operatorname{-mod}_{\lambda}^{B}\overset{\eqref{e:gmodline}}{\simeq}\operatorname{Vect}$

is an equivalence, i.e. is given by tensoring by a cohomologically graded
line. To see this, we may replace $\Gamma$ by $\mathring{\Gamma}$, and we
simply unwind, using the equivariance of the appearing functors and
adjunction:

$\displaystyle\operatorname{Hom}_{\mathfrak{g}\operatorname{-mod}_{\lambda}^{B}}(M_{\lambda},\hskip
2.84526pti^{!}\circ\mathring{\Gamma}\circ\operatorname{Av}^{B}_{*}\hskip
1.42262pt(\delta_{\psi}))$
$\displaystyle\simeq\operatorname{Hom}_{\mathfrak{g}\operatorname{-mod}_{\lambda}^{B}}(M_{\lambda},\hskip
2.84526pt\operatorname{Av}^{B}_{*}\circ\hskip
2.84526pti^{!}\circ\mathring{\Gamma}\hskip 1.42262pt(\delta_{\psi}))$
$\displaystyle\simeq\operatorname{Hom}_{\mathfrak{g}\operatorname{-mod}}(M_{\lambda},\hskip
2.84526pt\mathring{\Gamma}\hskip 1.42262pt(\delta_{\psi}))$
$\displaystyle\simeq\operatorname{Hom}_{\mathfrak{g}\operatorname{-mod}}(M_{\lambda},\hskip
2.84526pt\operatorname{ind}_{\mathfrak{n}^{-}}^{\mathfrak{g}}k_{\psi}).$
Writing $\mathfrak{b}$ for the Lie algebra of $B$, and noting that
$\operatorname{Res}_{\mathfrak{g}}^{\mathfrak{b}}\circ\operatorname{ind}_{\mathfrak{n}^{-}}^{\mathfrak{g}}k_{-\psi}\simeq
U(\mathfrak{b})$, we compute
$\displaystyle\simeq\operatorname{Hom}_{\mathfrak{b}\operatorname{-mod}}(k_{\lambda},\hskip
2.84526pt\operatorname{Res}_{\mathfrak{g}}^{\mathfrak{b}}\circ\operatorname{ind}_{\mathfrak{n}^{-}}^{\mathfrak{g}}k_{\psi})$
$\displaystyle\simeq\det(\mathfrak{b}^{*}[-1]),$

as desired. ∎

######  Remark 5.2.5.

Plainly, $\Gamma$ is given by global sections on $G/N^{-},\psi$. It admits a
left adjoint $\operatorname{Loc}$ given by localization on $G/N^{-},\psi$. One
can show both functors are $t$-exact. In particular, Theorem 5.2.4 is
equivalent to the statement that $\operatorname{Loc}$ restricts to an
equivalence $\mathfrak{g}\operatorname{-mod}_{\lambda}\simeq
D(G/N^{-},\psi)^{B\operatorname{-gen}}$.

######  Remark 5.2.6.

The geometric side of the localization theorem may be thought of as follows.
The finite group of Lie type $G(\mathbb{F}_{q})$ has a distinguished simple
$k$-representation, the Steinberg module, constructed as follows. Within the
Whittaker model of $k$-valued functions

$\operatorname{Fun}(G(\mathbb{F}_{q})/N^{-}(\mathbb{F}_{q}),\psi)$

there is a unique $B(\mathbb{F}_{q})$-invariant line. It transforms as the
sign representation of the Hecke algebra, and generates the Steinberg module
for the group. Using the formalism of Section 3, we produced a sheaf-theoretic
analogue, $D(G/N^{-},\psi)^{B\operatorname{-gen}}$. So, a paraphrase of
Theorem 5.2.4 is that ‘a maximally singular block in
$\mathfrak{g}\operatorname{-mod}$ is the same as the Steinberg module’.

#### 5.2.7.

We now slightly rewrite the target of the localization theorem, using the
following.

######  Definition 5.2.8.

For a $D(G)$-module $\mathscr{C}$, define its Steinberg–Whittaker invariants
as

$\mathscr{C}^{N^{-},\psi,\operatorname{s}}:=\operatorname{Hom}_{D(G)\operatorname{-mod}}(D(G/N^{-},\psi)^{B\operatorname{-gen}},\mathscr{C})$

From the tautological adjunction
$D(G/N^{-},\psi)^{B\operatorname{-gen}}\rightleftarrows D(G/N^{-},\psi),$ we
deduce for any $D(G)$-module $\mathscr{C}$ an adjunction

$\mathscr{C}^{N^{-},\psi,\operatorname{s}}\rightleftarrows\mathscr{C}^{N^{-},\psi},$

wherein the left adjoint is fully faithful. Using this, we will slightly
rewrite $D(G/N^{-},\psi)^{B\operatorname{-gen}}$. Namely, we will replace the
appearing $B\operatorname{-gen}$, which refers to the action by left
convolution, with Steinberg–Whittaker invariants for the action by right
convolution.

######  Proposition 5.2.9.

The inclusions into $D(G/N^{-},\psi)$ yield a $D(G)$-equivariant equivalence

$D(G/N^{-},\psi)^{B\operatorname{-gen}}\simeq
D(G/N^{-},\psi,\operatorname{s}).$

###### Proof.

For any $D(G)$-module $\mathscr{C}$, we have by Corollary 3.1.8

$\displaystyle\mathscr{C}^{N^{-},\psi,\operatorname{s}}$
$\displaystyle\simeq\operatorname{Hom}_{D(G)\operatorname{-mod}}(D(G/N^{-},\psi)^{B\operatorname{-gen}},\mathscr{C})$
$\displaystyle\simeq\operatorname{Hom}_{D(G)\operatorname{-mod}}(D(G/N^{-},\psi),\mathscr{C}^{B\operatorname{-gen}})\simeq
D(N^{-},\psi\backslash G/B)\underset{D(B\backslash
G/B)}{\otimes}\mathscr{C}^{B}.$

Applying this to $\mathscr{C}=D(G)$, viewed as a $D(G)$-module via convolution
on the right, yields the proposition. ∎

Before moving on, let us describe the simplest case in more detail.

######  Example 5.2.10.

Suppose that $G=T$ is a torus. In this case, we have explicitly that
$\mathscr{C}^{N^{-},\psi,\operatorname{s}}\simeq\mathscr{C}^{T\operatorname{-mon}},$
cf. Example 3.1.6 for the definition of the latter. In particular, Proposition
5.2.9 reduces to the observation

$D(T\operatorname{-mon}\backslash T)\simeq D(T/T\operatorname{-mon}),$

since both are the full subcategory of $D(T)$ compactly generated by the
constant sheaf.

###  5.3. The weakly equivariant realization

#### 5.3.1.

Recall the tautological identification of Lie algebra representations and
weakly equivariant D-modules on the group

$\Gamma^{G}:D(G)^{G,w}\simeq\mathfrak{g}\operatorname{-mod},$

cf. Section 2.5. In the remainder of this section, we identify the full
subcategory of $D(G)^{G,w}$ corresponding to
$\mathfrak{g}\operatorname{-mod}_{\lambda}$, and more generally do so for
$\mathfrak{g}\operatorname{-mod}_{\nu}$, for any $\nu\in\Lambda$.

To state the answer, recall that in Section 4.0.12 to any $D(G)$-module
$\mathscr{C}$ and character $\chi\in\Lambda$ we attached a certain full
subcategory of the weak invariants

$\mathscr{C}^{G,w,\chi}\subset\mathscr{C}^{G,w}.$

######  Proposition 5.3.2.

For any $\nu\in\Lambda$, the composition
$\mathfrak{g}\operatorname{-mod}_{\nu}\subset\mathfrak{g}\operatorname{-mod}\simeq
D(G)^{G,w}$ factors through a $D(G)$-equivariant equivalence

(5.7) $\mathfrak{g}\operatorname{-mod}_{\nu}\simeq D(G)^{G,w,-\nu-2\rho}.$

###### Proof.

Consider the commutative diagram

(5.8) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
59.83484pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{g}\operatorname{-mod}_{\nu}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
125.5365pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-21.20667pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\wr}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
125.5365pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{D(G)^{G,w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
18.1332pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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83.83484pt\raise-40.26056pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-63.87pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\wr}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
83.83484pt\raise-42.41333pt\hbox{\hbox{\kern 0.0pt\raise
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59.00153pt\raise-44.5661pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Hom}_{D(G)\operatorname{-mod}}(\mathfrak{g}\operatorname{-mod}_{\nu}^{\vee},D(G))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
83.83484pt\raise-83.17389pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
83.83484pt\raise-85.32666pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Hom}_{D(G)\operatorname{-mod}}(D(G)^{G,w},D(G))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
59.83484pt\raise-87.47943pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
125.5365pt\raise-127.48999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{}^{G,w}D(G)}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where to pass between the second and third rows we use for dualizable
$D(G)$-modules $\mathscr{M}$ and $\mathscr{N}$ the tautological equivalence
$\operatorname{Hom}_{D(G)\operatorname{-mod}}(\mathscr{M},\mathscr{N})\simeq\operatorname{Hom}_{D(G)\operatorname{-mod}}(\mathscr{N}^{\vee},\mathscr{M}^{\vee}).$

It is straightforward to see that the composition of the right column of
equivalences in (5.8) is the equivalence induced by inversion $g\mapsto
g^{-1}$ on the group

$\operatorname{inv}_{*}:D(G)^{G,w}\simeq{}^{G,w}D(G).$

Accordingly, to prove the proposition, it is enough to show that the standard
self-duality of $D(G)^{G,w}$, i.e. the one induced by Verdier duality and the
identification of weak invariants and coinvariants, induces an equivalence

(5.9)
$\mathfrak{g}\operatorname{-mod}_{\nu}^{\vee}\simeq\mathfrak{g}\operatorname{-mod}_{-\nu-2\rho}.$

Under $\Gamma^{G}$, the self-duality of $D(G)^{G,w}$ interchanges with the
self-duality of $\mathfrak{g}\operatorname{-mod}$ given by Lie algebra
cohomology, so we are done by Proposition 4.0.14. ∎

##  6\. parabolic induction for groups and Hecke categories

To prove the general case of the localization theorem, we will reduce to the
maximally singular case using parabolic induction for categorical
representations. In this section, we first collect basic definitions and
properties of parabolic induction functors for groups and Hecke algebras. We
then establish a basic compatibility between these induction functors, and
deduce a useful corollary on submodules generated by Borel-invariant vectors.

The reader mostly interested in representation-theoretic applications may wish
to skim Section 6.1 for the basic definitions and read the comments following
Corollary 6.3.3 before proceeding on to Section 7.

###  6.1. Definitions and first properties

#### 6.1.1.

Let $G$ and $B$ be as in Section 2. Fix a standard parabolic subgroup
$B\subset P\subset G$.888Note that for a loop group $G$, these are sometimes
called instead parahoric subgroups. Denote its prounipotent radical and Levi
quotient by $N_{P}$ and $L$ respectively. One then has the standard
correspondence of groups

$L\leftarrow P\rightarrow G.$

######  Definition 6.1.2.

The functor of parabolic induction is given by the composition

(6.1)
$\operatorname{pind}_{P}^{G}:D(L)\operatorname{-mod}\xrightarrow{\operatorname{res}_{L}^{P}}D(P)\operatorname{-mod}\xrightarrow{\operatorname{ind}_{P}^{G}}D(G)\operatorname{-mod}.$

By its definition, parabolic induction is left adjoint to the Jacquet, or
parabolic restriction  functor

(6.2)
$\operatorname{pres}_{G}^{P}:D(G)\operatorname{-mod}\xrightarrow{\operatorname{res}_{G}^{L}}D(P)\operatorname{-mod}\xrightarrow{\operatorname{inv}^{N_{P}}}D(L)\operatorname{-mod}.$

######  Remark 6.1.3.

Parabolic induction also right adjoint to the Jacquet functor, although in
affine type this depends on the choice of a dimension theory on $G$ to
normalize the equivalence $D(G)\simeq D(G)^{\vee}$, cf. Section 2.3.2.

When $P$ and $G$ are clear from context, we will refer to the functors simply
as $\operatorname{pind}$ and $\operatorname{pres}$. It is straightforward to
see that both functors are $\operatorname{DGCat}_{\operatorname{cont}}$-linear
and commute with limits and colimits. Similarly, $\operatorname{pres}$
canonically commutes with duality, and the same holds for
$\operatorname{pind}$ with the caveat for $G$ of affine type as in Remark
6.1.3.

#### 6.1.4.

Let us pass to Hecke categories. Writing $B_{L}\simeq B/N_{P}$ for the induced
Borel subgroup of $L$, one has a correspondence

$B_{L}\backslash L/B_{L}\leftarrow B\backslash P/B\rightarrow B\backslash
G/B.$

This induces a correspondence of monoidal functors

$D(B_{L}\backslash L/B_{L})\leftarrow D(B\backslash P/B)\rightarrow
D(B\backslash G/B),$

wherein the leftward arrow, by the prounipotence of $N_{P}$, is an
equivalence, and the rightward arrow, as it is a pushforward along a closed
embedding, is fully faithful.

######  Definition 6.1.5.

The functor of parabolic induction is the composition

(6.3)
$\operatorname{pind}_{\mathscr{H}_{L}}^{\mathscr{H}_{G}}:D(B_{L}\backslash
L/B_{L})\operatorname{-mod}\simeq D(B\backslash
P/B)\operatorname{-mod}\xrightarrow{\operatorname{ind}}D(B\backslash
G/B)\operatorname{-mod}.$

As before, we have that
$\operatorname{pind}_{\mathscr{H}_{L}}^{\mathscr{H}_{G}}$ is left adjoint to
the parabolic restriction functor

(6.4) $\operatorname{pres}_{\mathscr{H}_{G}}^{\mathscr{H}_{L}}:D(B\backslash
G/B)\xrightarrow{\operatorname{res}}D(B\backslash
P/B)\operatorname{-mod}\simeq D(B_{L}\backslash L/B_{L})\operatorname{-mod}.$

######  Remark 6.1.6.

It follows from Corollary 3.1.8 that
$\operatorname{pind}_{\mathscr{H}_{L}}^{\mathscr{H}_{G}}$ is also canonically
right adjoint to $\operatorname{pres}_{\mathscr{H}_{G}}^{\mathscr{H}_{L}}$.

When it is clear from context that we are referring to the Hecke categories
associated to $P$ and $G$, we will refer to the functors simply as
$\operatorname{pind}$ and $\operatorname{pres}$. It is straightforward to see
that $\operatorname{pind}$ and $\operatorname{pres}$ are
$\operatorname{DGCat}_{\operatorname{cont}}$-linear, and commute with limits,
colimits, and duality.

###  6.2. Commutation of parabolic induction and Borel invariants

#### 6.2.1.

Having now discussed parabolic induction functors for groups and Hecke
categories, we now establish their compatibility. Informally, the following
theorem states that taking Borel invariant vectors commutes with parabolic
induction.

######  Theorem 6.2.2.

The following diagram commutes up to natural equivalence:

(6.5) ${D(L)\operatorname{-mod}}$${D(B_{L}\backslash
L/B_{L})\operatorname{-mod}}$${D(G)\operatorname{-mod}}$${D(B\backslash
G/B)\operatorname{-mod}.}$$\scriptstyle{\operatorname{inv}^{B_{L}}}$$\scriptstyle{\operatorname{pind}_{L}^{G}}$$\scriptstyle{\operatorname{pind}_{\mathscr{H}_{L}}^{\mathscr{H}_{G}}}$$\scriptstyle{\operatorname{inv}^{B}}$

###### Proof.

Note that the appearing functors are
$\operatorname{DGCat}_{\operatorname{cont}}$-linear and commute with colimits.
In particular, the composition
$\operatorname{pind}_{\mathscr{H}_{L}}^{\mathscr{H}_{G}}\circ\operatorname{inv}^{B_{L}}$
is given by tensoring over $D(L)$ with the $D(B\backslash G/B)\otimes
D(L)$-module

$D(B\backslash G/B)\underset{D(B\backslash P/B)}{\otimes}D(B_{L}\backslash
L)\simeq D(B\backslash G/B)\underset{D(B\backslash P/B)}{\otimes}D(B\backslash
P/N_{P}).$

Similarly, the composition
$\operatorname{inv}^{B}\circ\operatorname{pind}_{L}^{G}$ is given by tensoring
over $D(L)$ with the $D(B\backslash G/B)\otimes D(L)$-module

$D(B\backslash G)\underset{D(G)}{\otimes}D(G/N_{P})\simeq D(B\backslash
G/N_{P}).$

Therefore, it is enough to show the natural convolution morphism

(6.6) $D(B\backslash G/B)\underset{D(B\backslash P/B)}{\otimes}D(B\backslash
P/N_{P})\rightarrow D(B\backslash G/N_{P}).$

is an equivalence. As this identifies with the counit map for the action of
$D(L)$ on $D(B\backslash G/N_{P})$ via right convolution, by Theorem 3.1.5 it
is fully faithful.

It remains to prove the essential surjectivity of (6.6). For this, let $W$
denote the Weyl group of $G$, and for each $w\in W$ consider the Schubert cell

$C_{w}:=BwB\subset G.$

Write $j_{w}:B\backslash C_{w}\rightarrow B\backslash G$ for the associated
locally closed embedding. In view of the Cousin filtration and the already
established fully faithfulness, it suffices to show that $j_{w,*}\mathscr{M}$
belongs to the essential image of (6.6) for any $w\in W$ and any $\mathscr{M}$
in $D(B\backslash C_{w}/N_{P})$.

In fact, we will obtain all such $\mathscr{M}$ by convolution from D-modules
on a single Schubert cell of $L$, as follows. Write $w=uy$ where $y\in W_{L}$
and $u$ is the minimal length representative of the coset $wW_{L}$. Then,
writing $\ell$ for the standard length function on the Coxeter group $W$,
$\ell(w)=\ell(u)+\ell(y)$ and convolution yields an isomorphism

(6.7) $C_{u}\stackrel{{\scriptstyle B}}{{\times}}C_{y}\longrightarrow C_{w}.$

In addition, consider the tautological projection

$\pi:C_{u}\stackrel{{\scriptstyle B}}{{\times}}C_{y}\rightarrow B\backslash
C_{y}.$

By our assumption on $u$ and $y$, this induces an isomorphism after
quotienting by $B$ and $N_{P}$, i.e.

(6.8) $\pi:B\backslash C_{u}\stackrel{{\scriptstyle
B}}{{\times}}C_{y}/N_{P}\simeq B\backslash C_{y}/N_{P}.$

Finally, note the resulting composition

(6.9) $D(B\backslash
C_{y}/N_{P})\overset{\eqref{e:step2}}{\simeq}D(B\backslash
C_{u}\stackrel{{\scriptstyle
B}}{{\times}}C_{y}/N_{P})\overset{\eqref{e:step1}}{\simeq}D(B\backslash
C_{w}/N_{P})\xrightarrow{j_{w,*}}D(B\backslash G/N_{P})$

is given by convolution on the left with the costandard object $j_{u,*}$. In
particular, the image of (6.9) lies in the image of (6.6), as desired. ∎

######  Remark 6.2.3.

By passing to the right adjoints of the functors appearing in (6.5), one
obtains a similar commutative diagram for parabolic restriction.

###  6.3. Compatibility with generation by Borel invariants

#### 6.3.1.

We now record a consequence of Theorem 6.2.2 which will be of use in what
follows. Recall that to a $D(L)$-module $\mathscr{C}$ or a $D(G)$-module
$\mathscr{D}$ one has the associated submodules generated by their Borel
invariants

$\mathscr{C}^{B_{L}\operatorname{-gen}}\quad\text{and}\quad\mathscr{D}^{B\operatorname{-gen}},$

respectively, cf. Section 3.2.1 for their definition.

#### 6.3.2.

The following corollary, said informally, asserts that the operation of taking
the submodule generated by Borel invariants commutes with parabolic induction
and restriction.

######  Corollary 6.3.3.

There is a natural isomorphism of functors $D(L)\operatorname{-mod}\rightarrow
D(G)\operatorname{-mod}$:

(6.10)
$\operatorname{pind}_{L}^{G}(\mathscr{C}^{B_{L}\operatorname{-gen}})\simeq(\operatorname{pind}_{L}^{G}\mathscr{C})^{B\operatorname{-gen}}.$

Similarly, there is a natural isomorphism of functors
$D(G)\operatorname{-mod}\rightarrow D(L)\operatorname{-mod}$:

(6.11)
$\operatorname{pres}_{G}^{L}(\mathscr{D}^{B\operatorname{-gen}})\simeq(\operatorname{pres}_{G}^{L}\mathscr{D})^{B_{L}\operatorname{-gen}}.$

###### Proof.

We begin with (6.10). Note that the left hand side of (6.10) is given by
tensoring over $D(L)$ with the $D(L)\otimes D(G)$-module

$D(G)\underset{D(P)}{\otimes}D(L/B_{L})\underset{D(B_{L}\backslash
L/B_{L})}{\otimes}D(B_{L}\backslash L)\simeq D(G/B)\underset{D(B_{L}\backslash
L/B_{L})}{\otimes}D(B_{L}\backslash L).$

Similarly, the right hand side is given by tensoring with

$\displaystyle D(G/B)\underset{D(B\backslash G/B)}{\otimes}D(B\backslash
G)\underset{D(G)}{\otimes}D(G/N_{P})$ $\displaystyle\simeq
D(G/B)\underset{D(B\backslash G/B)}{\otimes}D(B\backslash
G\stackrel{{\scriptstyle G}}{{\times}}G/N_{P})$ $\displaystyle\simeq
D(G/B)\underset{D(B\backslash G/B)}{\otimes}D(B\backslash G/N_{P}).$ To
identify the two, we may apply (6.6) to $D(B\backslash G/N_{P})$ to further
rewrite the right hand side as
$\displaystyle{\simeq}D(G/B)\underset{D(B\backslash
G/B)}{\otimes}D(B\backslash G/B)\underset{D(B_{L}\backslash
L/B_{L})}{\otimes}D(B_{L}\backslash L)$ $\displaystyle\simeq
D(G/B)\underset{D(B_{L}\backslash L/B_{L})}{\otimes}D(B_{L}\backslash L),$

as desired. The identification (6.11) now follows by either adjunction or a
similar analysis. ∎

Corollary 6.3.3 in particular implies that generation by Borel invariant
vectors, cf. Definition 3.2.2, survives parabolic induction and restriction.
So, we may control such categorical representations of groups via Hecke
categories. Let us now apply this to localization theory.

##  7\. Localization II: the parabolic induction of the Steinberg module and
Lie algebra representations

Let $G$ and $B$ be as in Section 4. Recall the set of integral weights
$\Lambda$, cf. Sections 2.2 and 2.5, and fix $\lambda\in\Lambda$ such that the
corresponding Verma module $M_{\lambda}$ is irreducible.

#### 7.0.1.

In this section, we prove localization theorems for the corresponding category

(7.1) $\mathfrak{g}\operatorname{-mod}_{\lambda}.$

We will do so by identifying this with the parabolic induction of a similar
category for a Levi. The latter will be, by construction, maximally singular.
Applying parabolic induction to its two geometric realizations from Section 5
will provide the desired localization of (7.1).

###  7.1. The parabolic induction of the Steinberg module

#### 7.1.1.

Note that, since $M_{\lambda}$ is irreducible, by the representation theory of
$\mathfrak{sl}_{2}$ it follows that the weight $\lambda$ is anti-dominant,
i.e. for any simple coroot $\check{\alpha}$ we have

(7.2) $\langle\check{\alpha},\lambda+\rho\rangle\leqslant 0.$

######  Remark 7.1.2.

In our present setting the reverse implication holds too, but this is a
nontrivial theorem of Kac–Kazhdan in the affine case (at non-critical level)
[KK79]. We emphasize that we do not need this fact.

In particular, the stabilizer of $\lambda$ under the dot action of $W$ is a
parabolic subgroup $W_{P}$ corresponding to a proper parabolic $P\supset B.$
As in Section 6, let us write $N_{P}$ for the prounipotent radical of $P$ and
$L$ for its Levi quotient.

#### 7.1.3.

Let us write $\mathfrak{l}$ for the Lie algebra of $L$, and consider the
corresponding $D(L)$-module

$\mathfrak{l}\operatorname{-mod}_{\lambda}.$

By construction, it is maximally singular, i.e. falls into the regime of
Section 5. As in Section 6, we may consider its parabolic induction from $L$
to $G$:

(7.3)
$\operatorname{pind}_{L}^{G}(\mathfrak{l}\operatorname{-mod}_{\lambda}).$

We now use the results of Section 5, along with a compatibility of Borel
invariants and parabolic induction established in Section 6, to rewrite (7.3)
geometrically. In the following proposition, we write $\rho_{L}$ for the half
sum of positive roots of $L$. In addition, for a generic character $\psi$ of
$N_{L}$, we denote by the same letter its inflation to $N$, i.e. the
composition

$N\rightarrow N/N_{P}\simeq N_{L}\xrightarrow{\psi}\mathbb{G}_{a}.$

In particular, for a $D(G)$-module $\mathscr{C}$, we may consider its
Steinberg–Whittaker invariants

$\mathscr{C}^{N,\psi,\operatorname{s}}:=(\mathscr{C}^{N_{P}})^{N_{L},\psi,\operatorname{s}},$

cf. Section 5.2.7 for the definition the Steinberg–Whittaker invariants of a
$D(L)$-module.

######  Example 7.1.4.

Note that if $L$ is the abstract Cartan $T$, the above definition simply
recovers the monodromic invariants, i.e.

$\mathscr{C}^{N,\psi,\operatorname{s}}\simeq\mathscr{C}^{B\operatorname{-mon}},$

cf. Example 3.1.6 for the definition of the latter.

######  Proposition 7.1.5.

There are canonical $D(G)$-equivariant equivalences

$\operatorname{pind}_{L}^{G}(\mathfrak{l}\operatorname{-mod}_{\lambda})\simeq
D(G/N,\psi,\operatorname{s})\simeq D(G/N_{P})^{L,w,-\lambda-2\rho_{L}}.$

###### Proof.

In Section 5, we saw $D(L)$-equivariant equivalences

$\mathfrak{l}\operatorname{-mod}_{\lambda}\simeq
D(L/N_{L},\psi,\operatorname{s})\simeq D(L)^{L,w,-\lambda-2\rho_{L}}.$

In particular, we may parabolically induce these to obtain equivalences

$\operatorname{pind}(\mathfrak{l}\operatorname{-mod}_{\lambda})\simeq\operatorname{pind}(D(L/N_{L},\psi,\operatorname{s}))\simeq\operatorname{pind}(D(L)^{L,w,-\lambda-2\rho_{L}}).$

To rewrite the middle term, we first note the
$\operatorname{DGCat}_{\operatorname{cont}}$-linearity and continuity of
$(N_{L},\psi,\operatorname{s})$ invariants. Indeed, these follow via
colocalization from the analogous assertions for $(N,\psi)$ invariants. Using
this observation, we may rewrite the middle term as

$\displaystyle\operatorname{pind}(D(L/N_{L},\psi,\operatorname{s}))$
$\displaystyle\simeq(\operatorname{pind}(D(L)))^{N_{L},\psi,\operatorname{s}}$
$\displaystyle\simeq(D(G)\underset{D(P)}{\otimes}D(P/N_{P}))^{N_{L},\psi,\operatorname{s}}$
$\displaystyle\simeq D(G/N_{P})^{N_{L},\psi,\operatorname{s}}$
$\displaystyle\simeq D(G/N,\psi,\operatorname{s}).$

Similarly, using the $\operatorname{DGCat}_{\operatorname{cont}}$-linearity
and continuity of $(L,w,-\lambda-2\rho_{L})$ invariants, which again follow
their analogues for $(L,w)$ invariants, we may rewrite the right hand term as

$\displaystyle\operatorname{pind}(D(L)^{L,w,-\lambda-2\rho_{L}})$
$\displaystyle\simeq
D(G)\underset{D(P)}{\otimes}{D(P/N_{P})^{L,w,-\lambda-2\rho_{L}}}$
$\displaystyle\simeq D(G/N_{P})^{L,w,-\lambda-2\rho_{L}},$

as desired. ∎

######  Remark 7.1.6.

Note that by Corollary 6.3.3 and Theorem 5.2.4, we have

$D(G/N,\psi,\operatorname{s})\simeq D(G/N,\psi)^{B\operatorname{-gen}}.$

#### 7.1.7.

Let us use Proposition 7.1.5 to obtain a collection of compact generators for

(7.4)
$\operatorname{pind}_{L}^{G}(\mathfrak{l}\operatorname{-mod}_{\lambda})^{B}.$

To do so, write $\mathscr{M}_{e}$ for the image of the Verma module
$M_{\lambda}^{L}$ in $\mathfrak{l}\operatorname{-mod}_{\lambda}^{B_{L}}$ under
the unit of adjunction

$\mathfrak{l}\operatorname{-mod}_{\lambda}^{B_{L}}\rightarrow\operatorname{pind}_{L}^{G}(\mathfrak{l}\operatorname{-mod}_{\lambda})^{B}.$

Recall that $j_{w,!}\star M_{\lambda}^{L}\simeq M_{\lambda}^{L}$ for $w\in
W_{P}$, cf. Lemma 4.0.8. In particular, we may consider the objects

$\mathscr{M}_{w}:=j_{w,!}\star\mathscr{M}_{e},\quad\quad\text{for }w\in
W/W_{P}.$

######  Corollary 7.1.8.

The objects $\mathscr{M}_{w}$, for $w\in W/W_{P}$, compactly generate (7.4).
Moreover, we have the vanishing

(7.5)
$\operatorname{Hom}(\mathscr{M}_{w},\mathscr{M}_{e})\simeq\begin{cases}k&\text{if
}w=e,\\\ 0&\text{otherwise.}\end{cases}$

###### Proof.

Proposition 7.1.5 and Remark 7.1.6 yield an identification

$\operatorname{pind}_{L}^{G}(\mathfrak{l}\operatorname{-mod}_{\lambda})^{B}\simeq
D(B\backslash G/N,\psi).$

By construction this identifies $\mathscr{M}_{e}$ with the simple object of
$D(B\backslash P/N,\psi)$. In particular the $\mathscr{M}_{w}$, for $w\in
W/W_{P}$, are identified with the standard objects of $D(B\backslash
G/N,\psi)$, which makes the claims manifest. ∎

###  7.2. The parabolic induction of the Steinberg module and Lie algebra
representations

#### 7.2.1.

To prove the localization theorems for
$\mathfrak{g}\operatorname{-mod}_{\lambda}$, it is enough to identify it with
$\operatorname{pind}_{L}^{G}(\mathfrak{l}\operatorname{-mod}_{\lambda})$. We
now observe that there is a canonical functor in one direction.

#### 7.2.2.

Indeed, note that parabolic induction of Lie algebra representations is a
$D(L)$-equivariant functor

(7.6)
$\operatorname{pind}_{\mathfrak{l}}^{\mathfrak{g}}:\mathfrak{l}\operatorname{-mod}\rightarrow\mathfrak{g}\operatorname{-mod}^{N_{P}},$

cf. Section 2.5.4 for the equivariance data.

######  Remark 7.2.3.

In the affine case, if $\lambda$ is a weight of level $\kappa$, by this we
mean the functor of level $\kappa$ parabolic induction

$\mathfrak{l}\operatorname{-mod}\rightarrow\widehat{\mathfrak{h}}_{\kappa}\operatorname{-mod}^{N_{P}}\hskip
2.84526pt\subset\hskip 2.84526pt\mathfrak{g}\operatorname{-mod}.$

By adjunction, (7.6) induces a $D(G)$-equivariant functor

$\operatorname{pind}_{L}^{G}(\mathfrak{l}\operatorname{-mod})\rightarrow\mathfrak{g}\operatorname{-mod}.$

As parabolic induction sends Verma modules to Verma modules, this restricts to
a $D(G)$-equivariant functor

(7.7)
$\operatorname{pind}_{L}^{G}(\mathfrak{l}\operatorname{-mod}_{\lambda})\rightarrow\mathfrak{g}\operatorname{-mod}_{\lambda}.$

#### 7.2.4.

We now show that the above yields the sought-for identification.

######  Theorem 7.2.5.

The functor (7.7) is an equivalence.

###### Proof.

As both sides of (7.7) are generated by their Borel invariants (cf. Corollary
6.3.3 for the left hand side), it is enough to show the obtained
$D(B\backslash G/B)$-equivariant functor

$\gamma:\operatorname{pind}_{L}^{G}(\mathfrak{l}\operatorname{-mod}_{\lambda})^{B}\rightarrow\mathfrak{g}\operatorname{-mod}_{\lambda}^{B}$

is an equivalence.

We first claim it admits an equivariant right adjoint $\gamma^{R}$. Indeed,
note that for any $w\in W/W_{P}$ we have

(7.8) $\gamma(\mathscr{M}_{w})\simeq\gamma(j_{w,!}\star\mathscr{M}_{e})\simeq
j_{w,!}\star\gamma(\mathscr{M}_{e})\simeq j_{w,!}\star
M_{\lambda}\overset{\eqref{e:stb1}}{\simeq}M_{w\cdot\lambda}.$

In particular, $\gamma$ sends compact objects to compact objects, and hence
admits a continuous right adjoint, which is equivariant by the semi-rigidity
of $D(B\backslash G/B)$.

We next check that $\gamma$ is fully faithful. As $\mathscr{M}_{e}$ generates
(7.4) as a $D(B\backslash G/B)$-module, it is enough to see the canonical map

(7.9) $\mathscr{M}_{e}\rightarrow\gamma^{R}\circ\gamma(\mathscr{M}_{e})$

is an equivalence. Indeed, for any $w\in W/W_{P}$, we unwind

(7.10)
$\displaystyle\operatorname{Hom}(\mathscr{M}_{w},\gamma^{R}\circ\gamma(\mathscr{M}_{e}))$
$\displaystyle\simeq\operatorname{Hom}(\gamma(\mathscr{M}_{w}),\gamma(\mathscr{M}_{e}))\overset{\eqref{e:mgoestom}}{\simeq}\operatorname{Hom}(M_{w\cdot\lambda},M_{\lambda}).$
Recalling that $M_{\lambda}$ is simple, and in particular coincides with its
contragredient dual $A_{\lambda}$, we may continue (7.11)
$\displaystyle\simeq\operatorname{Hom}(M_{w\cdot\lambda},A_{\lambda})\simeq\begin{cases}k&\text{if
}w=e\\\ 0&\text{otherwise;}\end{cases}$

where the last vanishing is a (standard) consequence of the cofreeness of
$A_{\lambda}$ over $N$. Comparing with (7.5), it follows that (7.9) is an
equivalence, as desired.

It remains to see that $\gamma$ is essentially surjective. However, as the
$M_{w\cdot\lambda}$ compactly generate the target, cf. Proposition 4.0.7, we
are done by (7.8). ∎

#### 7.2.6.

Concatenating Theorem 7.2.5 and Proposition 7.1.5, we have shown the
following.

######  Theorem 7.2.7.

There are canonical $D(G)$-equivariant equivalences

(7.12) $\mathfrak{g}\operatorname{-mod}_{\lambda}\simeq
D(G/N,\psi,\operatorname{s})\simeq D(G/N_{P})^{L,w,-\lambda-2\rho_{L}}.$

#### 7.2.8.

Let us explain how to deduce localization theorems at positive level as well.

######  Corollary 7.2.9.

There are canonical $D(G)$-equivariant equivalences

$\mathfrak{g}\operatorname{-mod}_{-\lambda-2\rho}\simeq
D(G/N,-\psi,\operatorname{s})\simeq D(G/N_{P})^{L,w,\lambda}.$

###### Proof.

Recall that in Proposition 4.0.14 we saw that

$\mathfrak{g}\operatorname{-mod}_{\lambda}^{\vee}\simeq\mathfrak{g}\operatorname{-mod}_{-\lambda-2\rho}.$

The corollary therefore follows by dualizing the categories appearing in
(7.12). ∎

######  Remark 7.2.10.

Consider the action of $T$ (supplemented with the loop rotation
$\mathbb{G}_{m}$ in affine type, i.e. the Levi quotient of the automorphisms
of a formal disk) on the additive characters of $N$. As $\psi$ and $-\psi$ lie
in the same orbit, one may choose an identification

$D(G/N,\psi,\operatorname{s})\cong D(G/N,-\psi,\operatorname{s}).$

The resulting equivalence
$\mathfrak{g}\operatorname{-mod}_{\lambda}\cong\mathfrak{g}\operatorname{-mod}_{-\lambda-2\rho}$
behaves as a ‘long intertwining operator’.

##  8\. The linkage principle for positive energy representations

The goal of this section is to prove a linkage principle for
$\mathfrak{g}\operatorname{-mod}^{B\operatorname{-gen}}$ and specialize it to
a linkage principle for positive energy representations proposed by Yakimov in
[Yak07].

The proofs of this section do not rely on localization theory, but do use the
definitions and results of Sections 3, 4, and 6. The only material from this
section used in its sequels is Theorem 8.1.6, which will be applied to obtain
the block decomposition for affine Harish-Chandra bimodules. So, the reader
may wish to examine Theorem 8.1.6 and then proceed directly to Section 9.

###  8.1. Positive energy representations

#### 8.1.1.

Let $H$ be an almost simple, simply connected algebraic group, $H[[z]]$ its
arc group, and $H(\\!(z)\\!)$ its loop group. Passing to their Lie algebras,
let us write $\mathfrak{h}$ for the corresponding semisimple Lie algebra,
$\mathfrak{h}[[z]]$ for the corresponding arc algebra, and
$\widehat{\mathfrak{h}}_{\kappa}$ for the corresponding affine Lie algebra at
a fixed noncritical integral level. Recall that $\Lambda_{f}$ denotes the
character lattice of the abstract Cartan of $H$.

#### 8.1.2.