# The Drinfeld–Sokolov reduction of admissible representations of affine Lie
algebras

Gurbir Dhillon  Yale University
New Haven, CT 06511, USA<EMAIL_ADDRESS>
###### Abstract.

Fix an affine Lie algebra $\widehat{\mathfrak{g}}_{\kappa}$ with associated
principal affine W-algebra $\mathcal{W}_{\kappa}$. A basic conjecture of
Frenkel–Kac–Wakimoto asserts that Drinfeld–Sokolov reduction sends admissible
$\widehat{\mathfrak{g}}_{\kappa}$-modules to zero or cohomological shifts of
minimal series $\mathcal{W}_{\kappa}$-modules. We prove this conjecture and a
natural generalization to the spectrally flowed Drinfeld–Sokolov reduction
functors. This extends the previous results of Arakawa for the minus
reduction.

##  1\. Introduction

### 1.1.

In the seminal work [FF90a] Feigin and Frenkel introduced the affine
W-algebras into mathematics. These algebras and their representation theory
have been intensively studied in the ensuing three decades and have
fundamental relations to integrable hierarchies, moduli spaces associated to
algebraic curves, and the geometric Langlands program, cf. [Ara17], [FBZ04],
[Fre07a] and references therein.

In this paper we settle a basic conjecture from [FF90a], or more precisely a
refinement due to Frenkel–Kac–Wakimoto [FKW92]. In the remainder of the
introduction we review some of the history and context of the conjecture; the
reader may wish to skip directly to Section 2 where we state our results.

### 1.2.

The method by which Feigin and Frenkel produced the W-algebras is
cohomological in nature, and takes as its input a simple Lie algebra
$\mathfrak{g}$ equipped with an adjoint-invariant bilinear form $\kappa$. Let
us write $\widehat{\mathfrak{g}}_{\kappa}$ for the affine Lie algebra
associated to $\mathfrak{g}$ and $\kappa$, which is a central extension of the
loop algebra $\mathfrak{g}(\\!(t)\\!)$:

$0\rightarrow
k\cdot\mathbf{1}\rightarrow\widehat{\mathfrak{g}}_{\kappa}\rightarrow\mathfrak{g}(\\!(t)\\!)\rightarrow
0.$

By the construction of the associated W-algebra, which we denote by
$\mathcal{W}_{\kappa}$, one has a functor of Drinfeld–Sokolov reduction
between the derived categories of modules

$\Psi^{+}:\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\rightarrow\mathcal{W}_{\kappa}\operatorname{-mod}.$
(1.1)

Let us mention two basic properties of this functor. First, it sends the
vacuum Kac–Moody vertex algebra $\mathbb{V}_{\kappa}$, concentrated in
cohomological degree zero, to $\mathcal{W}_{\kappa}$, concentrated in
cohomological degree zero. Second, it is constructed via conformal field
theoretic techniques. As a consequence, the functor factorizes. That is, for
any smooth algebraic curve $X$, Equation (1.1) naturally extends to a functor
between the versions of the appearing categories of chiral algebra modules
defined over the Ran space of $X$, cf. [BD04].

### 1.3.

In the final pages of [FF90a], Feigin and Frenkel proposed a conjectural
property of Drinfeld–Sokolov reduction. Kac and Wakimoto had discovered for
any positive rational level $\kappa$ a family of irreducible highest weight
$\widehat{\mathfrak{g}}_{\kappa}$-modules, the admissible modules,
distinguished by the modularity properties of their characters [KW88], [KW89].
Feigin and Frenkel predicted that Drinfeld–Sokolov reduction should send
admissible modules of $\widehat{\mathfrak{g}}_{\kappa}$ to zero or irreducible
minimal series representations of $\mathcal{W}_{\kappa}$. The latter had
attracted interest in both mathematics and physics due to their appearance in
rational conformal field theory, cf. [BS93].

In particular, by an analysis of Euler characteristics, combined with the
character formula for admissible modules due to Kac–Wakimoto, the conjecture
would imply formulae for the $q$-characters of the minimal series, which was
the subject of forthcoming joint work of Frenkel–Kac–Wakimoto [FKW92].

######  Example 1.3.1.

Let us specialize the preceding discussion to the simplest case of
$\mathfrak{g}=\mathfrak{sl}_{2}$. Here, the algebra $\mathcal{W}_{\kappa}$ is
a Virasoro vacuum algebra, whose central charge varies with $\kappa$, and its
minimal series are the celebrated minimal models of chiral conformal field
theory [BPZ84]. As Feigin and Frenkel emphasize in [FF90a], the correspondence
with admissible $\widehat{\mathfrak{sl}}_{2}$-modules had already been
proposed and studied in the physical literature [KPZ88], [MP90], and proven by
the authors themselves in [FF90b].

### 1.4.

In the classic paper [FKW92], Frenkel–Kac–Wakimoto performed a comprehensive
analysis of the resulting $q$-characters and formulated a series of
conjectures based on their findings. Combining these with the Verlinde
formula, they obtained a conjectural calculation of the fusion rules for the
minimal series. We now recall two important features of their work.

A first basic discovery of Frenkel–Kac–Wakimoto was that, beyond the case of
$\mathfrak{g}=\mathfrak{sl}_{2}$, Drinfeld–Sokolov reduction could not simply
send an admissible module, concentrated in cohomological degree zero, to a
minimal series representation, concentrated in cohomological degree zero.
Instead, to account for certain signs encountered in their analysis of
characters, they conjectured that admissible modules are sent to zero or
cohomological shifts of minimal series. Moreover, they gave for each
admissible module a prediction for whether its reduction vanished, and if not
the exact cohomological shift and the highest weight for the obtained minimal
series representation. These conjectures intricately depend on the level
$\kappa$ and the highest weight of the admissible module. They have been since
regarded as the definitive formulation of the prediction announced by
Feigin–Frenkel on Drinfeld–Sokolov reduction.

A second basic contribution of Frenkel–Kac–Wakimoto was their introduction of
another functor

$\Psi^{-}:\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\rightarrow\mathcal{W}_{\kappa}\operatorname{-mod},$
(1.2)

which they called the minus Drinfeld–Sokolov reduction. In the case of
$\mathfrak{g}=\mathfrak{sl}_{2}$, this had been previously considered by
Feigin–Frenkel [FF90b]. This functor, which is a variant of the usual, i.e.
plus, Drinfeld–Sokolov reduction, is less immediate to construct using
conformal field theoretic techniques. Relatedly, it no longer exchanges the
vacuum vertex algebras, and does not extend to a factorizable functor.

However, a crucial realization of Frenkel–Kac–Wakimoto was that the minus
reduction should again send admissible modules to zero or minimal series, and
in a drastically simpler fashion than the plus reduction. More precisely, they
conjectured that the reduction of an admissible series representation would
not vanish if and only if its highest weight, considered as a weight of the
abstract Cartan of the finite dimensional algebra $\mathfrak{g}$, was
antidominant. They further conjectured that the resulting
$\mathcal{W}_{\kappa}$-module would always be a minimal series representation,
concentrated in cohomological degree zero.

The findings of Frenkel–Kac–Wakimoto, and in particular their conjectures on
the plus and minus reductions, have proven highly influential. They continue
to guide developments in the subject, e.g. the emerging connections between
minimal models and the quantum Langlands correspondence. We note that for the
latter connection, and more generally questions of a global nature involving
conformal blocks, the factorizability of the plus reduction plays an important
role.

### 1.5.

After both conjectures lay open for over a decade, a breakthrough was made by
Arakawa. In the tour-de-force works [Ara04], [Ara07], he completely proved the
conjecture for the minus reduction, and verified many cases of the conjecture
for the plus reduction. His work moreover yielded character formulae for every
simple highest weight $\mathcal{W}_{\kappa}$-module, extending the formulae
for minimal series conjectured by Frenkel–Kac–Wakimoto. We remark that an
alternative proof of the conjecture for the minus reduction was given by the
author and Raskin recently in the works [Ras21], [DR20].

### 1.6.

The decade following Arakawa’s work has seen tremendous progress in the study
of W-algebras. Notably, many of the main conjectures from the early days of
the subject, e.g. the relation with coset models and the rationality of
minimal quotients, have been fully resolved [Ara15], [ACL19]. However, to our
knowledge, no further cases of Frenkel–Kac–Wakimoto’s conjecture on the plus
reduction were obtained.111However, some cases of the analogous assertions for
other spectrally flowed reduction functors may be found in the interesting
work [ACF20].

The basic added difficulty for the conjecture on plus reduction can be
understood immediately from its statement. Namely, the absence of
cohomological shifts in the conjecture for the minus reduction underlies the
fact that the functor is $t$-exact on all of the Category $\mathscr{O}$ of
$\widehat{\mathfrak{g}}_{\kappa}$-modules. This was known for
$\mathfrak{g}=\mathfrak{sl}_{2}$ by the work of Feigin–Frenkel [FF90b], and
was proven in general by Arakawa [Ara04]. This $t$-exactness plays a crucial
role in the argument of Arakawa for the minus reduction, and that of the
author and Raskin. An analogous $t$-exactness holds for the plus reduction
applied to certain blocks of Category $\mathscr{O}$, which were exactly the
cases settled by Arakawa. Beyond these cases, the plus reduction is not
$t$-exact, so one needs a further idea to proceed.

### 1.7.

In this paper, we prove Frenkel–Kac–Wakimoto’s conjecture for the plus
reduction, as well as a natural generalization to arbitrary dominant weights
and spectrally flowed reduction functors.

##  2\. Statement of Results

In this section, after introducing the necessary notation and preliminaries,
we state our main Theorem 2.5.1.

### 2.1.

Fix an arbitrary level $\kappa$, and let $L_{\lambda}$ be an irreducible
highest weight $\widehat{\mathfrak{g}}_{\kappa}$-module which is dominant.
That is, we suppose its Verma cover $M_{\lambda}$ is projective in the
Category $\mathscr{O}$. In particular $L_{\lambda}$ could be any admissible
module.

### 2.2.

Fix in addition an element $\check{\mu}$ of the coweight lattice of the finite
Cartan. One may associate to it the spectrally flowed Drinfeld–Sokolov
reduction functor

$\Psi^{\check{\mu}}:\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\rightarrow\mathcal{W}_{\kappa}\operatorname{-mod},$
(2.1)

cf. Section 4 for the definition, which is slightly nonstandard if
$\check{\mu}$ is not antidominant. For now, it suffices to say that if we
write $\check{\rho}$ for one half of the sum of the positive finite coroots,
then the functor $\Psi^{\check{\rho}}$ equals the plus reduction, and the
functor $\Psi^{0}$ equals the minus reduction.

### 2.3.

Our main theorem calculates the spectrally flowed reductions

$\Psi^{\check{\mu}}(L_{\lambda})\in\mathcal{W}_{\kappa}\operatorname{-mod}.$

If we write $G$ for the simple algebraic group of adjoint type with Lie
algebra $\mathfrak{g}$, and $LG$ for its algebraic loop group, then as we
explain in Section 4 this calculation is equivalent to computing, for all
$g\in LG$, the plus reduction of the $\operatorname{Ad}_{g}$-conjugated simple
modules

$\Psi^{+}\big{(}\operatorname{Ad}_{g}(L_{\lambda})\big{)}.$

The answer involves the following Coxeter-theoretic combinatorics.

Let us denote the finite Weyl group, the extended affine Weyl group, and the
integral Weyl group of $L_{\lambda}$ by $W_{f},W,$ and $W_{\lambda},$
respectively. In particular, we have the tautological inclusions

$W_{f}\subset W\supset W_{\lambda}.$

Write $e^{\check{\mu}}$ for the element of $W$ corresponding to $\check{\mu}$.
Let us say that the pair $(\check{\mu},\lambda)$ is good if the double coset

$W_{f}e^{\check{\mu}}W_{\lambda}$ (2.2)

is a $W_{f}\times W_{\lambda}$ torsor, i.e. the action is simply transitive.
In this case one has a unique factorization

$e^{\check{\mu}}=w_{f}w_{-}w_{\lambda},$ (2.3)

where $w_{f}\in W_{f}$, $w_{-}$ is the unique minimal length element of (2.2)
with respect to the Bruhat order on $W$, and $w_{\lambda}\in W_{\lambda}$. We
write $\ell_{\lambda}(w_{\lambda})$ for the length of $w_{\lambda}$ viewed as
an element of the Coxeter system $W_{\lambda}$.

### 2.4.

Finally, we need some notation. Write $\mathfrak{t}^{*}$ for the dual of the
finite Cartan. Recall that the combinatorics of highest weight
$\widehat{\mathfrak{g}}_{\kappa}$-modules is governed by the level $\kappa$
dot action of $W$ on $\mathfrak{t}^{*}$, which we denote by

$W\times\mathfrak{t}^{*}\rightarrow\mathfrak{t}^{*},\quad\quad(w,\lambda)\mapsto
w\lambda,$

cf. Section 3.5.2 for further discussion of this action. To discuss highest
weights for $\mathcal{W}_{\kappa}$-modules, we denote the Harish–Chandra
projection by

$\pi:\mathfrak{t}^{*}\rightarrow\mathfrak{t}^{*}//W_{f},$

and for an infinitesimal character $\chi\in\mathfrak{t}^{*}//W_{f}$ we write
$L_{\chi}$ for the corresponding simple highest weight
$\mathcal{W}_{\kappa}$-module, cf. Section 3.5.4 for our precise
normalization.

### 2.5.

With these preparations, we can state our main result.

######  Theorem 2.5.1.

Let $\check{\mu}$ and $\lambda$ be as in Sections 2.2 and 2.1, respectively.
If the pair $(\check{\mu},\lambda)$ is good, let $w_{-}$ and $w_{\lambda}$ be
as in Equation (2.3). Then the corresponding spectrally flowed reduction is
given by

$\Psi^{\check{\mu}}(L_{\lambda})\simeq\begin{cases}L_{\pi(w_{-}\lambda)}[\ell_{\lambda}(w_{\lambda})]&\text{if
$(\check{\mu},\lambda)$ is good},\\\ 0&\text{otherwise.}\end{cases}$

We would like to make two comments about the theorem.

######  Remark 2.5.2.

Let us take $\check{\mu}=\check{\rho}$, so that we are considering the plus
reduction, and let us further suppose that $\lambda$ is a principal admissible
weight in the sense of Kac–Wakimoto [KW89]. This case of Theorem 2.5.1
confirms Conjecture $\operatorname{3.4}_{+}$ of Frenkel–Kac–Wakimoto [FKW92].

More precisely, (i) whether the reduction vanishes is exactly as they
conjectured. Further, (ii) when the reduction is nonvanishing we get the
irreducible $\mathcal{W}_{\kappa}$-module that they conjectured concentrated
in a single cohomological degree. Finally, (iii) the exact cohomological
degree corrects their conjecture. Namely, we find a shift by the length of
$w_{\lambda}$ rather than minus its semi-infinite length; these differ in
examples already for $\mathfrak{g}=\mathfrak{sl}_{3}$. Note however that this
does not affect the parity of the cohomological degree, and in particular is
consistent with their Euler characteristic calculations.

######  Remark 2.5.3.

After stating Conjecture $\operatorname{3.4}_{+}$, Frenkel–Kac–Wakimoto
outline in Conjectures 3.5.1 and 3.5.2 a conjecture on the cohomology of the
plus reduction of a 2-sided resolution of $L_{\lambda}$ by Wakimoto modules.
Their Conjecture 3.5.1 on the existence of two sided resolutions was settled
by Arakawa [Ara14], and Conjecture 3.5.2 then follows by Theorem 2.5.1.

### 2.6.

The method by which we prove Theorem 2.5.1, which we will indicate in the next
subsection, also yields tight bounds on the cohomological amplitude of any
$\Psi^{\check{\mu}}$ when restricted to a block of Category $\mathscr{O}$. The
right exactness of the functors follows from work of Frenkel–Gaitsgory [FG06].
Our lower bounds on the nonvanishing cohomology are to the best of our
knowledge new save for the case of the minus reduction, where they reduce to
its $t$-exactness.

For example, for the block containing $L_{\lambda}$ as above, we show the
functor has cohomological amplitude at most
$[-\ell_{\lambda}(w_{\lambda}),0]$. That is, the reduction of $L_{\lambda}$,
when nonzero, is concentrated in the most negative possible degree. The case
of a general block is similar and is presented in Theorem 5.5.1 below.

### 2.7.

Finally, let us describe the idea of the proof. Our starting point is the fact
that the various reductions $\Psi^{\check{\mu}}$, when restricted to Category
$\mathscr{O}$, differ from the minus reduction $\Psi^{-}$ by precomposition
with intertwining, i.e. twisting, functors.222The twisting functors were
constructed geometrically by Feigin–Frenkel in their fundamental work [FF90c],
and algebraically by Arkhipov in [Ark04]. They also appear in the earlier work
of Beilinson–Bernstein [BB83] under the name intertwining functors.
Equivalently, writing $\mathring{I}$ for the prounipotent radical of $I$, they
differ by convolution with costandard objects in the universal affine Hecke
category

$D_{\kappa}(\mathring{I}\backslash LG/\mathring{I}).$ (2.4)

Here universal refers to the fact that we impose no restrictions on the
monodromies of the D-modules along the orbits of the finite Cartan
$I/\mathring{I}$ under left and right multiplication. This relation with
intertwining functors is an old observation going back at least to the work of
Frenkel–Gaitsgory [FG06] (and, after applying the sheaves-functions
dictionary, far earlier still to the study of principal series representations
of finite and $p$-adic Chevalley groups).

In our approach to the conjecture we make essential use of the fact that, upon
restriction to a fixed block $\mathscr{O}_{\chi}$ of Category $\mathscr{O}$,
the relevant intertwining functors descend to costandard objects in an affine
Hecke category with right monodromy fixed by $\chi$. As such, this carries an
action by the bi $\chi$-monodromic affine Hecke category

$D_{\kappa}(\mathring{I}\backslash LG/I,\chi)\circlearrowleft
D_{\kappa}(I,\chi\backslash LG/I,\chi).$ (2.5)

This action categorifies the right action of the integral Weyl group on the
extended affine Weyl group

$W\circlearrowleft W_{\chi}.$

Practically, this means that the way in which costandard objects in (2.5)
convolve with one another depends finely upon $\chi$. The upshot for us is
that these convolutions satisfy identities which do not descend from the
universal case (2.4).

To prove Theorem 2.5.1 we lift the factorization

$e^{\check{\mu}}=w_{f}\circ w_{-}\circ w_{\chi}$

of Equation (2.3) to such an identity of costandard objects in (2.5). That is,
we write in Section 4 a spectrally flowed reduction functor as the minus
reduction precomposed with two intertwining functors:

$\Psi^{\check{\mu}}\simeq\Psi^{-}\circ j_{w_{-},*}^{w_{-}\chi,\chi}\circ
j_{w_{\chi},*}^{\chi,\chi}.$ (2.6)

We then track in Section 5 how each step in the factorization transforms
dominant highest weight modules in $\mathscr{O}_{\chi}$ and find that it
produces the behavior anticipated by Frenkel–Kac–Wakimoto. Namely, the
functors $\Psi^{-}$ and $j_{w_{-},*}^{w_{-}\chi,\chi}$ of Equation (2.6) are
$t$-exact and behave simply on the abelian categories of modules; for
$\Psi^{-}$ this is exactly Arakawa’s theorem. However, the first term
$j_{w_{\chi},*}^{\chi,\chi}$ is typically far from $t$-exact, and it is in
general difficult to calculate its action on a given object. However, on
‘maximally integrable’ objects, such as dominant highest weight modules, it
reduces to a cohomological shift of the identity functor, which yields the
theorem.

######  Remark 2.7.1.

The cases of the plus reduction previously obtained by Arakawa [Ara07] are
exactly the cases when $w_{\chi}$ is length zero. The latter condition is
equivalent to $j_{w_{\chi},*}^{\chi,\chi}$ being $t$-exact, and for this
reason the arguments for the minus reduction, where one checks that
(co)standard objects are sent to (co)standard objects, and deduce an assertion
for arbitrary simples, carry over mutatis mutandis.

In short, we find the combinatorics of the Frenkel–Kac–Wakimoto conjecture for
the plus reduction in the combinatorics of twisted affine Hecke categories.

### 2.8.

The paper is organized as follows. We gather in Section 3 some preliminary
material on loop groups, W-algebras, categorical representation theory,
Whittaker sheaves, and Hecke categories.

In Section 4 we relate the spectrally flowed reduction functors to Whittaker
sheaves on the affine flag manifold and obtain the factorization of Equation
(2.6).

Finally, in Section 5 we apply this to obtain the main Theorem 2.5.1.

Acknowledgments. We warmly thank Sam Raskin for conversations and
collaborations related to $\mathcal{W}_{\kappa}$-algebras, and in particular
for introducing us to the conjecture addressed in this paper several years
ago. We additionally thank him and Ivan Losev for comments on a draft of this
paper, and Zhiwei Yun for some helpful conversations.

We thank Tomoyuki Arakawa, Thomas Creutzig, Edward Frenkel, and Geordie
Williamson for correspondence related to the present work.

The author was supported by an NSF Postdoctoral Fellowship under grant No.
2103387.

##  3\. Notation and preliminaries

In this section we gather some known results and set notation. Aside from our
convention that our fixed Borel $B$ is $TN^{-}$, and the opposite unipotent
subgroup is $N$, our choices are more or less standard. So, the reader may
wish to skip directly to Section 4 and refer back only as needed.

###  3.1. Loop groups and algebras

Some references for the following material are [FBZ04], [Fre07b].

#### 3.1.1.

We work throughout over an algebraically closed field $k$ of characteristic
zero. In particular, ind-schemes and their morphisms discussed below are over
the spectrum of $k$. For an ind-scheme $X$ we denote by $X(k)$ its set of
$k$-points.

#### 3.1.2.

Let $G$ be a simple algebraic group of adjoint type.444The results of the
paper for the W-algebra associated to a general connected reductive group
follow immediately from this case. We fix a Cartan subgroup $T$ and Borel
subgroup $B=TN^{-}$, and write $N$ for the opposite maximal unipotent
subgroup. We denote the respective Lie algebras by
$\mathfrak{g},\mathfrak{t},\mathfrak{b},\mathfrak{n}^{-},\text{ and
}\mathfrak{n}.$ Finally, we write $N_{G}(T)$ for the normalizer of $T$ in $G$,
and denote the finite Weyl group by

$W_{f}:=N_{G}(T)/T.$

Recall this is a Coxeter system, i.e. a Coxeter group with a distinguished
collection of simple reflections, which in this case is fixed by $B$. For an
element $w\in W_{f}$, by a coset representative $\dot{w}$ we mean any lift to
$N_{G}(T)$.

#### 3.1.3.

Let us turn to the loop group. For an affine scheme $X$, we denote by $L^{+}X$
its formal arc space, which is a scheme of infinite type, and we denote by
$LX$ its formal loop space, which is an ind-scheme of infinite type. Recall
that there is a canonical closed embedding $L^{+}X\rightarrow LX$. As the
formation of arc and loop spaces is functorial and commutes with finite
products, they send affine group schemes to group ind-schemes.

Within the loop group $LG$, our choice of $B\subset G$ distinguishes an
Iwahori subgroup $I\subset LG$. Namely, $I$ is the preimage of $B$ under the
evaluation map $L^{+}G\rightarrow G.$

#### 3.1.4.

We write $N_{LG}(LT)$ for the normalizer of $LT$ in $LG$, and denote the
extended affine Weyl group by

$W:=N_{LG}(LT)/L^{+}T.$

For an element $w\in W$, by a coset representative $\dot{w}$ we mean any lift
to $N_{LG}(LT)$. If we denote the cocharacter lattice of $T$, i.e. the
coweight lattice, by $\check{\Lambda}$, we have a canonical isomorphism

$W\simeq W_{f}\ltimes\check{\Lambda}.$ (3.1)

Since there are different sign conventions for this isomorphism in the
literature, let us spell this out carefully. For a cocharacter
$\check{\mu}:\mathbb{G}_{m}\rightarrow T$, its restriction to the formal
punctured disk about $0\in\mathbb{A}^{1}(k)$ yields an element
$t^{\check{\mu}}\in LT(k)$.

With this, for an element $w\in W_{f}$, equipped with any lift $\dot{w}$ to
$N_{G}(T)$, we identify the $L^{+}T$ coset of $\dot{w}t^{\check{\mu}}$ on the
left-hand side of (3.1) with the element $we^{\check{\mu}}$ on the right-hand
side.

Let us also recall that if we write $\check{Q}$ for the coroot lattice, then
the subgroup $W_{f}\ltimes\check{Q}$ is canonically a Coxeter system, with its
simple reflections fixed by our Iwahori subgroup $I$. Its length function and
Bruhat order naturally extend to all of $W$, and we denote these by $\ell$ and
$<$ respectively.

#### 3.1.5.

Let us now discuss the loop algebras. Note that if $H$ is an affine algebraic
group with Lie algebra $\mathfrak{h}$, then the topological Lie algebras of
$L^{+}H$ and $LH$ are given by (the $k$-points of) $L^{+}\mathfrak{h}$ and
$L\mathfrak{h}$, respectively. Explicitly, one has canonical isomorphisms

$L^{+}\mathfrak{h}\simeq\mathfrak{h}\otimes k[[t]]\quad\text{and}\quad
L\mathfrak{h}\simeq\mathfrak{h}\otimes k(\\!(t)\\!),$

with the pointwise Lie bracket.

#### 3.1.6.

For a level, i.e. Ad-invariant bilinear form $\kappa$ on $\mathfrak{g}$, we
denote the associated central extension of its loop algebra by

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

and the associated vacuum vertex algebra by

$\mathbb{V}_{\kappa}\simeq\operatorname{ind}_{L^{+}\mathfrak{g}\oplus
k\cdot\mathbf{1}}^{\widehat{\mathfrak{g}}_{\kappa}}k.$

We would like to next review the definition of the associated affine
W-algebra.

###  3.2. Semi-infinite cohomology and the $\mathcal{W}_{\kappa}$-algebra

The definition of the affine W-algebra is due to Feigin–Frenkel [FF90a]. Some
references for the following material are [Vor93], [FBZ04], and [Ara17].

#### 3.2.1.

In this paper, we only need the following simple case of semi-infinite
cohomology. Suppose $U$ is a unipotent algebraic group with Lie algebra
$\mathfrak{u}$. For a smooth $L\mathfrak{u}$-module $M$ and a compact open
subalgebra $\mathfrak{k}$ of $L\mathfrak{u}$, one may form the semi-infinite
cohomology

$C^{\frac{\infty}{2}+*}(L\mathfrak{u},\mathfrak{k},M)\in\operatorname{Vect},$

where $\operatorname{Vect}$ denotes the dg-category of unbounded complexes of
$k$-vector spaces.555To set up some of the formalism we use and to construct
some of the appearing categories, one has to work systematically with dg-
categories. For this reason, we work with the canonical dg-enhancement of the
derived category of vector spaces. However, for the purposes of this paper,
one only really needs the existence of the associated triangulated categories
throughout, so the reader may safely ignore such points.

Recall that $\mathfrak{k}$ appears to trivialize the dimension torsor for
compact open subalgebras. Plainly, for another compact open subalgebra
$\tilde{\mathfrak{k}}$ one has a canonical isomorphism

$C^{\frac{\infty}{2}+*}(L\mathfrak{u},\tilde{\mathfrak{k}},M)\otimes\operatorname{rel.det}(\tilde{\mathfrak{k}},\mathfrak{k})^{*}\simeq
C^{\frac{\infty}{2}+*}(L\mathfrak{u},\mathfrak{k},M),$

where in the appearing relative determinant line both $\mathfrak{k}$ and
$\tilde{\mathfrak{k}}$ lie in cohomological degree -1.

#### 3.2.2.

Fix a generic additive character $\psi^{-}:N^{-}\rightarrow\mathbb{G}_{a}$.
Associated to our coordinate $t$ is an induced additive character

$LN^{-}\xrightarrow{L\psi^{-}}L\mathbb{G}_{a}\xrightarrow{\operatorname{Res}(-)}\mathbb{G}_{a},$
(3.2)

where, at the level of $k$-points, $\operatorname{Res}(-)$ sends a formal
power series

$f(t)\in k(\\!(t)\\!)\simeq L\mathbb{G}_{a}(k)$

to its coefficient of $t^{-1}$. We again use $\psi^{-}$ to denote the
composition (3.2), and we write $k_{\psi^{-}}$ for the induced one dimensional
representations of $\mathfrak{n}^{-}$ and $L\mathfrak{n}^{-}$. With this, by
definition the underlying vector space of $\mathcal{W}_{\kappa}$ is the
complex

$C^{\frac{\infty}{2}+*}(L\mathfrak{n}^{-},L^{+}\mathfrak{n}^{-},\mathbb{V}_{\kappa}\otimes
k_{\psi^{-}}),$

which is concentrated cohomologically in degree zero.

###  3.3. Categories of smooth modules

#### 3.3.1.

For a dg-category $\mathscr{C}$ equipped with a $t$-structure we denote by
$\mathscr{C}^{+}$ the full subcategory of bounded below objects. Similarly,
for an integer $n$ we denote by $\mathscr{C}^{\leqslant n}$ and
$\mathscr{C}^{\geqslant n}$ the full subcategories of objects concentrated in
degrees at most $n$ and at least $n$, respectively. We denote the heart of the
$t$-structure, i.e. the abelian category of objects in cohomological degree
zero, by $\mathscr{C}^{\heartsuit}$.

#### 3.3.2.

For the affine Lie algebra, consider its abelian category
$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{\heartsuit}$ of smooth
modules on which $\mathbf{1}$ acts via the identity. We denote by
$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}$ its associated
renormalized derived category, as introduced by Frenkel–Gaitsgory [FG09]. We
recall that this carries a $t$-structure and a canonical $t$-exact functor to
the naive unbounded derived category

$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\rightarrow\operatorname{D}(\hskip
1.42262pt\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{\heartsuit}),$

which is an equivalence on bounded below categories. So, the only difference
between the two lies in cohomological degree $-\infty$.

#### 3.3.3.

For the affine W-algebra, consider its abelian category
$\mathcal{W}_{\kappa}\operatorname{-mod}^{\heartsuit}$ of smooth modules. We
denote by $\mathcal{W}_{\kappa}\operatorname{-mod}$ its renormalized derived
category, as introduced by Raskin [Ras21]. Again, one has a $t$-exact functor

$\mathcal{W}_{\kappa}\operatorname{-mod}\rightarrow\operatorname{D}(\mathcal{W}_{\kappa}\operatorname{-mod}^{\heartsuit})$

which induces an equivalence of the categories of bounded below objects.

###  3.4. The plus and minus reductions

The plus reduction below was introduced by Feigin–Frenkel in [FF90a]. The
minus reduction was studied for $\mathfrak{sl}_{2}$ by Feigin–Frenkel [FF90b],
and was introduced in general by Frenkel–Kac–Wakimoto [FKW92].

#### 3.4.1.

From the definition of the affine W-algebra, semi-infinite cohomology defines
a functor

$C^{\frac{\infty}{2}+*}\big{(}L\mathfrak{n}^{-},L^{+}\mathfrak{n}^{-},(-)\otimes
k_{\psi}^{-}\hskip
1.42262pt\big{)}:\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\rightarrow\mathcal{W}_{\kappa}\operatorname{-mod}.$
(3.3)

On bounded below objects, the composition

$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{+}\rightarrow\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\rightarrow\mathcal{W}_{\kappa}\operatorname{-mod}\rightarrow\operatorname{D}(\mathcal{W}_{\kappa}\operatorname{-mod}^{\heartsuit})$

agrees with standard construction via Feigin’s semi-infinite cochain complex
[Fei84]. Following Frenkel–Kac–Wakimoto, we call (3.3) the plus reduction. We
denote it by $\Psi^{+}$.

#### 3.4.2.

Fix in addition a generic additive character $\psi$ of the opposite unipotent
subalgebra $N$. Associated to our coordinate $t$ is an additive character

$LN\xrightarrow{L\psi}L\mathbb{G}_{a}\xrightarrow{\operatorname{Res}(t\cdot-)}\mathbb{G}_{a},$

where, at the level of $k$-points, $\operatorname{Res}(t\cdot-)$ sends a
formal power series to its coefficient of $1$. We denote by $k_{\psi}$ the
associated one dimensional representation of $L\mathfrak{n}$.

Let us write $L^{(1)}\mathfrak{n}$ for the first congruence subalgebra of
$L^{+}\mathfrak{n}$, so that

$L^{(1)}\mathfrak{n}\simeq\mathfrak{n}\otimes t\cdot k[[t]].$

Consider the associated semi-infinite cohomology.

$C^{\frac{\infty}{2}+*}\big{(}L\mathfrak{n},L^{(1)}\mathfrak{n},(-)\otimes
k_{\psi}\big{)}:\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\rightarrow\operatorname{Vect}.$
(3.4)

#### 3.4.3.

The functor (3.4) lifts to a functor to
$\mathcal{W}_{\kappa}\operatorname{-mod}$ as follows. Recalling that $G$ is
adjoint, it follows there is a unique coset representative $\dot{w}_{\circ}$
for the longest element $w_{\circ}$ of $W_{f}$ satisfying

$\operatorname{Ad}_{\dot{w}_{\circ}}(\psi)=\psi^{-},$

i.e. that exchanges the additive characters of $N$ and $N^{-}$ under
conjugation.

Write $\check{\rho}$ for one half of the sum of the positive coroots of $G$.
Passing to the loop groups, it follows that the element
$x:=\dot{w}_{\circ}t^{\check{\rho}}$ satisfies

$\operatorname{Ad}_{x}(\psi)=\psi^{-},$

i.e. it exchanges the additive characters of $LN$ and $LN^{-}$ under
conjugation.

Using this and the canonical isomorphism

$\displaystyle
C^{\frac{\infty}{2}+*}\big{(}L\mathfrak{n},L^{(1)}\mathfrak{n},(-)\otimes
k_{\psi}\big{)}$ $\displaystyle\simeq
C^{\frac{\infty}{2}+*}\big{(}L\mathfrak{n}^{-},\operatorname{Ad}_{x}L^{(1)}\mathfrak{n},\operatorname{Ad}_{x}(-)\otimes
k_{\psi^{-}}\hskip 1.42262pt\big{)}$ $\displaystyle\simeq
C^{\frac{\infty}{2}+*}\big{(}L\mathfrak{n}^{-},L^{+}\mathfrak{n}^{-},\operatorname{Ad}_{x}(-)\otimes
k_{\psi^{-}}\hskip 1.42262pt\big{)}\otimes\ell,$ where $\ell$ is the
cohomologically graded line $\displaystyle\ell$
$\displaystyle\simeq\operatorname{rel.det}(\operatorname{Ad}_{x}L^{(1)}\mathfrak{n},L^{+}\mathfrak{n}^{-}),$

we thereby obtain a functor

$C^{\frac{\infty}{2}+*}\big{(}L\mathfrak{n},L^{(1)}\mathfrak{n},(-)\otimes
k_{\psi}\big{)}:\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\rightarrow\mathcal{W}_{\kappa}\operatorname{-mod}.$

Following Frenkel–Kac–Wakimoto, we call this the minus reduction. We denote it
by $\Psi^{-}$.

###  3.5. Highest weight modules and the minus reduction

#### 3.5.1.

Recall that we fixed a Borel subgroup $B$ of $G$, and that we write $I$ for
the associated Iwahori subgroup of $LG$. For an element
$\lambda\in\mathfrak{t}^{*}$, we denote the corresponding one dimensional
representation of $\mathfrak{t}$ by $k_{\lambda}$ and the corresponding Verma
module of $\widehat{\mathfrak{g}}_{\kappa}$ by

$M_{\lambda}:=\operatorname{pind}_{\mathfrak{t}}^{\widehat{\mathfrak{g}}_{\kappa}}k_{\lambda}.$

We denote the simple quotient of $M_{\lambda}$ by $L_{\lambda}$.

#### 3.5.2.

Recall that the combinatorics of such highest modules are controlled by the
level $\kappa$ dot action of $W$ on $\mathfrak{t}^{*}$, which we denote by

$W\times\mathfrak{t}^{*}\rightarrow\mathfrak{t}^{*},\quad\quad(w,\mu)\mapsto
w\mu.$

Explicitly, $W_{f}$ acts by the usual dot action of the finite Weyl group, and
a coweight $\check{\mu}\in\check{\Lambda}$ acts by translation by

$(\kappa_{c}-\kappa)(\check{\mu})\in\mathfrak{t}^{*},$

where $\kappa_{c}$ denotes the critical level. In this formula, we have used
the restriction of $(\kappa_{c}-\kappa)$ from $\mathfrak{g}$ to
$\mathfrak{t}$, and the induced map

$(\kappa_{c}-\kappa):\mathfrak{t}\rightarrow\mathfrak{t}^{*}.$

#### 3.5.3.

We also recall that to any $\lambda\in\mathfrak{t}^{*}$ one associates its
integral Weyl group $W_{\lambda}$. To define this, write $\check{\Phi}^{+}$
for the positive real affine coroots, i.e. the positive real coroots of
$\widehat{\mathfrak{g}}_{\kappa}$. For $\check{\alpha}\in\check{\Phi}^{+}$,
consider the dual real root $\alpha$ of $\widehat{\mathfrak{g}}_{\kappa}$, and
write $\alpha_{c}$ for its classical part, i.e. associated root of
$\mathfrak{g}$. There is unique affine linear functional on
$\mathfrak{t}^{*}$, which we denote again by $\check{\alpha}$, such that the
corresponding reflection $s_{\check{\alpha}}$ in $W$ acts through the level
$\kappa$ dot action by

$s_{\check{\alpha}}\mu=\mu-\langle\mu,\check{\alpha}\rangle\alpha_{c}.$

With this, we may associate to $\lambda$ its positive integral coroots

$\check{\Phi}_{\lambda}^{+}:=\\{\check{\alpha}\in\check{\Phi}^{+}:\langle\check{\alpha},\lambda\rangle\in\mathbb{Z}\\}.$

By definition, $W_{\lambda}$ is the subgroup of $W$ generated by the
reflections $s_{\check{\alpha}},$ for
$\check{\alpha}\in\check{\Phi}^{+}_{\lambda}$. Moreover, if we call an element
$\check{\alpha}\in\check{\Phi}^{+}_{\lambda}$ indecomposable if it cannot be
written as a sum of two other elements of $\check{\Phi}^{+}_{\lambda}$, then
the reflections in $W_{\lambda}$ associated to the indecomposable integral
coroots exhibit $W_{\lambda}$ as a Coxeter system. Note that any simple
reflection of $W$ contained in $W_{\lambda}$ is necessarily simple in
$W_{\lambda}$, but a simple reflection of $W_{\lambda}$ need not be simple in
$W$.

#### 3.5.4.

We next review the associated categories of highest weight modules. Let us
write $\Lambda$ for the the character lattice of $T$, i.e. the root lattice.
Recall the canonical bijection between character sheaves on $T$ and
$\mathfrak{t}^{*}/\Lambda$, which associates to a character sheaf the
monodromy of the associated local system.

For a point $\chi\in\mathfrak{t}^{*}/\Lambda$, we denote by the same letter
the associated character sheaf on $I$, and by

$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}$

the category of $(I,\chi)$-equivariant
$\widehat{\mathfrak{g}}_{\kappa}$-modules. Plainly, this is compactly
generated the Verma modules

$M_{\lambda},\quad\text{for }\lambda\in\chi+\Lambda,$

and is a renormalization of the corresponding piece of (cocompleted, derived)
Category $\mathscr{O}$. For more discussion, the reader may see for example
[Gai15], Section 5 of [CD19], or Section 9 of [Ras20].

#### 3.5.5.

We would like to recall the behavior of the minus reduction on
$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}$. To do so, let
us write $\mathfrak{t}^{*}//W_{f}$ for the invariant theory quotient of
$\mathfrak{t}^{*}$ by the dot action of $W_{f}$. We normalize its
identification with the spectrum of the Zhu algebra

$\operatorname{Spec}\operatorname{Zhu}(\mathcal{W}_{\kappa})\simeq\mathfrak{t}^{*}//W_{f}$

as in [Dhi21]. For an infinitesimal character
$\Theta\in\mathfrak{t}^{*}//W_{f}(k)$, we denote the corresponding Verma
module for $\mathcal{W}_{\kappa}$ by $M_{\Theta}$ and its simple quotient by
$L_{\Theta}$.

The following fundamental theorem of Arakawa settled the Frenkel–Kac–Wakimoto
conjecture for the minus reduction [Ara04], [Ara07], for an alternative proof
cf. [Ras21], [DR20]. To state it, denote the Harish-Chandra projection by

$\pi:\mathfrak{t}^{*}\rightarrow\mathfrak{t}^{*}//W_{f},$

and recall that a weight $\mu\in\mathfrak{t}^{*}$ is said to be finite
antidominant if for every finite positive coroot $\check{\alpha}$ one has

$\langle\check{\alpha},\lambda\rangle\notin\mathbb{Z}^{>0},$

where we incorporate a $\rho$-shift into the pairing as in Section 3.5.3.

######  Theorem 3.5.6.

The minus reduction

$\Psi^{-}:\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}\rightarrow\mathcal{W}_{\kappa}\operatorname{-mod}$

is $t$-exact. Moreover, for any $\lambda\in\chi+\Lambda$ there are canonical
isomorphisms $\Psi^{-}(M_{\lambda})\simeq M_{\pi(\lambda)},$ and

$\Psi^{-}(L_{\lambda})\simeq\begin{cases}L_{\pi(\lambda)}&\text{if $\lambda$
is finite antidominant},\\\ 0&\text{otherwise.}\end{cases}$

That is, the minus reduction sends Verma modules to Verma modules and simple
modules to simple modules or zero.

###  3.6. Categorical actions of loop groups

#### 3.6.1.

We find it convenient and clarifying to use some language and tools from the
categorical representation theory of loop groups in what follows. However, the
reader may straightforwardly translate our arguments to the setting of [FG06]
(which, along with [BD], laid the groundwork for this theory).

Some references for the ensuing material are as follows. For generalities on
dg-categories, one may consult Section 1 of [GR17]. For D-modules on ind-
schemes, one may look at [Ras15] or the user-friendly [Ber17]. For the basic
results in categorical representation theory, one may see [Ber17], [Ras20],
and [Gai20], as well as the notes [ABC+18].

#### 3.6.2.

Consider the $(\infty,2)$-category of cocomplete, presentable $k$-linear dg-
categories $\operatorname{DGCat}_{\operatorname{cont}}$. We refer to objects
of it as dg-categories, and 1-morphisms in it, i.e. $k$-linear quasi-functors
in the sense of [Dri04] which commute with colimits, as functors.

$\operatorname{DGCat}_{\operatorname{cont}}$ is a symmetric monoidal
$\infty$-category with respect to the Lurie tensor product.666Using the
identification between the arising internal Homs and the morphisms in
$\operatorname{DGCat}_{\operatorname{cont}}$, one can alternatively work only
with the $(\infty,1)$-category associated to
$\operatorname{DGCat}_{\operatorname{cont}}$, i.e. discard non-invertible
2-morphisms. The monoidal unit is the category $\operatorname{Vect}$. We
denote the underlying binary product of two dg-categories $\mathscr{C}$ and
$\mathscr{D}$ by $\mathscr{C}\otimes\mathscr{D}$, and for objects $c$ of
$\mathscr{C}$ and $d$ of $\mathscr{D}$ we denote by $c\boxtimes d$ the
corresponding ‘external product’ object of $\mathscr{C}\otimes\mathscr{D}$.

Suppose $\mathscr{C}$ and $\mathscr{D}$ are compactly generated, which is the
case for all dg-categories considered in this paper. Concretely,
$\mathscr{C}\otimes\mathscr{D}$ is then compactly generated by external
products of compact objects from each factor, and the complexes of
homomorphisms between such objects are simply the tensor product

$\operatorname{Hom}_{\mathscr{C}\otimes\mathscr{D}}(c_{1}\boxtimes
d_{1},c_{2}\boxtimes
d_{2})\simeq\operatorname{Hom}_{\mathscr{C}}(c_{1},c_{2})\otimes\operatorname{Hom}_{\mathscr{D}}(d_{1},d_{2})$

with the natural compositions.

The category of $\kappa$-twisted D-modules on the loop group, which we denote
by $D_{\kappa}(LG)$, is naturally an algebra object of
$\operatorname{DGCat}_{\operatorname{cont}}$. The underlying binary product is
given by convolution, i.e. pushforward along the multiplication map

$LG\times LG\rightarrow LG.$

We denote its $(\infty,2)$-category of left modules by
$D_{\kappa}(LG)\operatorname{-mod}$. Plainly, an object of
$D_{\kappa}(LG)\operatorname{-mod}$ is a dg-category $\mathscr{C}$ equipped
with an action

$D_{\kappa}(LG)\otimes\mathscr{C}\rightarrow\mathscr{C}$

which is unital and associative up to coherent homotopy, and a 1-morphism
between representations is a functor between the underlying dg-categories
equipped with a datum of $D_{\kappa}(LG)$-equivariance.

The main example of interest for us is the canonical $D_{\kappa}(LG)$-module
structure on $\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}$, see for
example [Ras20]. To get a feel for this action, we note that for an element
$g\in LG$, convolution with the corresponding delta function

$\delta_{g}\star-:\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\simeq\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}$

is induced by the adjoint action of $g$ on $\widehat{\mathfrak{g}}_{\kappa}$.

###  3.7. Invariant vectors and representations

Given a $D_{\kappa}(LG)$-module $\mathscr{C}$, one can form its categories of
twisted Iwahori and Whittaker invariant objects

$\mathscr{C}^{I,\chi}\quad\text{and}\quad\mathscr{C}^{LN,\psi}.$

In our example of interest, these will exactly be the highest weight modules
$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}$ of Section
3.5.4, and the category $\mathcal{W}_{\kappa}\operatorname{-mod}$ of Section
3.3.3, respectively. We now review this in more detail.

#### 3.7.1.

We begin with the general construction of invariants. So, consider an ind-
affine placid group ind-scheme $H$ equipped with a character sheaf $\phi$, see
e.g. [Ras15] for a discussion of placidity.

######  Remark 3.7.2.

The following technical point may be safely ignored by the reader. The
category of D-modules on a placid ind-scheme $X$ which we denote by $D(X)$ is
denoted in [Ras15] and [Ber17] by $D^{*}(X)$, and comes with functoriality for
$*$-pushforwards. There is a dual category $D^{!}(X)$ which comes with
functoriality for $!$-pullbacks. The latter is symmetric monoidal under
$!$-tensor product, and canonically acts on $D^{*}(X)$ by $!$-tensor product.
In particular, when we speak of a character sheaf $\phi$, it canonically is an
object of $D^{!}(X)$. However, a choice of a dimension theory on $X$, which
always exists in our cases of interest, identifies $D^{!}(X)$ and $D(X)$.

Associated to the character sheaf $\phi$ is a $D(H)$-module
$\operatorname{Vect}_{\phi}$. Explicitly, its underlying dg-category is
$\operatorname{Vect}$, and the action map is of the form

$D(H)\otimes\operatorname{Vect}_{\phi}\rightarrow\operatorname{Vect}_{\phi},\quad\quad
M\boxtimes
V\mapsto\Gamma_{\operatorname{dR}}(H,M\overset{!}{\otimes}\phi)\otimes V,$
(3.5)

where in the right-hand expression $\overset{!}{\otimes}$ is the $!$-tensor
product $D(X)\otimes D^{!}(X)\rightarrow D(X)$, cf. Remark 3.7.2, and the
outer tensor product is of $k$-vector spaces. For any $D(H)$-module
$\mathscr{C}$, one may accordingly form its invariants and coinvariants

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

Recalling the tautological equivalence

$D(H)\otimes\operatorname{Vect}_{\phi}\simeq D(H),$

the action map (3.5) underlies a $D(H)$-equivariant functor
$D(H)\rightarrow\operatorname{Vect}_{\phi}$, which induces canonical
‘forgetting’ and ‘inserting’ maps

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

#### 3.7.3.

Suppose further now that $H$ is a group scheme. In this case, the map
$D(H)\rightarrow\operatorname{Vect}_{\phi}$ admits a left adjoint (i.e.,
$D(H)$ contains a constant sheaf) and hence one obtains an ‘averaging’
adjunction

$\operatorname{Oblv}:\mathscr{C}^{H,\phi}\leftrightarrows\mathscr{C}:\operatorname{Av}_{*}^{H,\phi}.$

Moreover, $\operatorname{Av}_{*}^{H,\phi}$ canonically factors through through
$\operatorname{ins}$, and if the prounipotent radical of $H$ is of finite
codimension the resulting map is an equivalence

$\mathscr{C}_{H,\phi}\xrightarrow{\sim}\mathscr{C}^{H,\phi}.$

We freely pass between invariants and coinvariants for such groups in what
follows.

Let us take $H$ to be the Iwahori $I$. The twisting $\kappa$ is canonically
trivialized on $L^{+}G$, and we use the induced trivialization on its subgroup
$I$ going forwards. In particular, any $D_{\kappa}(LG)$-module $\mathscr{C}$
carries an action of $D(I)$. With this, for an element
$\chi\in\mathfrak{t}^{*}/\Lambda$ we may form the twisted invariants
$\mathscr{C}^{I,\chi}$. Specializing to

$\mathscr{C}=\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod},$

the obtained category
$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}$ canonically
identifies with the renormalized derived category of Harish-Chandra modules
discussed in Section 3.5.4, cf. the references therein for the details. We
note in passing that a similar statement holds for any compact open subgroup
$H$ of $LG$.

#### 3.7.4.

The contents of Section 3.7.3 admit an analogue for the Whittaker invariants,
despite $LN$ being an ind-scheme. Recall the generic additive character

$\psi:LN\rightarrow\mathbb{G}_{a}$

of Section 3.4.2. We denote by the same letter the character sheaf
$\psi^{!}(\operatorname{exp}(z))$, where $\operatorname{exp}(z)$ is the
character sheaf on $\mathbb{G}_{a}$ whose underlying D-module is the
exponential D-module placed in cohomological degree -1.

The twisting $\kappa$ is canonically trivial on $LN$, and so to a
$D_{\kappa}(LG)$-module $\mathscr{C}$ one may form its Whittaker
(co)invariants. It will be convenient for us to use the canonical equivalence

$\mathscr{C}_{LN,\psi}\simeq\mathscr{C}^{LN,\psi},$

conjectured by Gaitsgory, and proven by Beraldo for $GL_{n}$ and in general by
Raskin [Ber17], [Ras21].

We denote by $\operatorname{Av}_{*}^{\psi,r}$ the renormalized averaging
functor, i.e. the composition

$\mathscr{C}\xrightarrow{\operatorname{ins}}\mathscr{C}_{LN,\psi}\simeq\mathscr{C}^{LN,\psi}.$

It is given by convolution with an object, which we call the (twisted)
renormalized constant sheaf

$k^{\psi,r}\in D(LN)^{LN,\psi}\subset D_{\kappa}(LG)^{LN,\psi}.$

#### 3.7.5.

Let us recall an explicit presentation of $k^{\psi,r}$. To do so, first
consider any ind-scheme of the form

$A\simeq\varinjlim_{\alpha}A_{\alpha},$

where each $A_{\alpha}$ is a pro-finite dimensional affine space and each
transition map $A_{\alpha}\hookrightarrow A_{\beta}$ is the closed embedding
of an open affine subspace of finite codimension (here, the word open means
with respect to the pro-topology on $A_{\beta}$). Each $A_{\alpha}$, as with
any placid scheme, carries a canonical constant sheaf $k(A_{\alpha})$. Namely,
consider any presentation of $A_{\alpha}$ as a cofiltered limit

$A_{\alpha}\simeq\varprojlim_{\gamma}A_{\alpha}^{\gamma},$

where the $A_{\alpha}^{\gamma}$ are finite dimensional affine spaces, and the
transition maps are smooth surjections. It follows that the $*$-pushforwards
$D(A_{\alpha}^{\gamma})\rightarrow D(A_{\alpha}^{\gamma^{\prime}})$ admit left
adjoints, i.e. $*$-pullbacks. We have by definition that

$D(A_{\alpha})\simeq\varinjlim_{\gamma}D(A_{\alpha}^{\gamma}),$

i.e. its category of D-modules is the colimit in
$\operatorname{DGCat}_{\operatorname{cont}}$ of those on its finite
dimensional quotients under $*$-pullback. In particular, the insertion of the
constant sheaf from any step in the colimit is the sought-for constant sheaf
$k(A_{\alpha})$.

For any fixed index $\alpha_{\circ}$, there is an associated renormalized
constant sheaf $k(A,A_{\alpha_{\circ}})$ on all of $A$. To describe this, note
that by our assumptions the $*$-pushforwards $D(A_{\alpha})\rightarrow
D(A_{\beta})$ admit right adjoints, i.e. $!$-pullbacks. With this,
$k(A,A_{\alpha_{\circ}})$ is determined uniquely up to unique isomorphism by
the properties that its $!$-restriction to any $A_{\beta}$ is isomorphic to a
cohomological shift of a constant sheaf, and its $!$-restriction to
$A_{\alpha_{\circ}}$ is $k(A_{\alpha_{\circ}})$. Explicitly, we have by
definition

$D(A)\simeq\varprojlim_{\alpha}D(A_{\alpha}),$

i.e. its category of D-modules is the cofiltered inverse limit in
$\operatorname{DGCat}_{\operatorname{cont}}$ of those on open profinite
dimensional subspaces under $!$-pullback. Hence, the existence follows from
the fact that such $!$-pullbacks send, by our assumption on the maps
$A_{\alpha}\hookrightarrow A_{\beta}$, constants sheaves to shifts of constant
sheaves.

######  Remark 3.7.6.

As the notation suggests, the objects $k(A,A_{\alpha_{\circ}})$ are semi-
infinite in nature, appearing like a constant sheaf along the pro-directions
and a dualizing sheaf along the ind-directions. Informally, they may be
thought of as halfway between the dualizing sheaf and the constant sheaf,
which do not exist as objects of $D(A)$. In particular, the various
$k(A,A_{\alpha})$ agree up to tensoring by cohomologically graded lines.

Let us apply this discussion to $A=LN$. We then have, in the conventions of
the present paper, that the kernel for renormalized averaging is

$k^{\psi,r}\simeq k(LN,L^{+}N)\overset{!}{\otimes}\psi.$ (3.6)

#### 3.7.7.

It will also be convenient to use the affine Skryabin equivalence

$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{LN,\psi}\simeq\mathcal{W}_{\kappa}\operatorname{-mod},$

which is an important theorem of Raskin [Ras21]. We will normalize this by
asking that it exchange the functor $\operatorname{Av}_{*}^{\psi,r}$ and the
semi-infinite cohomology

$C^{\frac{\infty}{2}+*}\big{(}L\mathfrak{n},L^{+}\mathfrak{n},(-)\otimes
k_{\psi}\big{)}:\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\rightarrow\mathcal{W}_{\kappa}\operatorname{-mod}.$

######  Remark 3.7.8.

In the alternative proof of Arakawa’s Theorem 3.5.6 by the author and Raskin,
the affine Skryabin equivalence plays a central role. Here, modulo the facts
about the minus reduction, we will only use that semi-infinite cohomology
factors through the Whittaker coinvariants, which is more or less automatic.

###  3.8. Convolution algebras

#### 3.8.1.

Suppose that $H$ is as in Section 3.7.1, and is equipped with an embedding
$H\hookrightarrow LG$ and a trivialization $\tau$ of the restriction of
$\kappa$. In this case, by tensor-hom adjunction we have for any
$D_{\kappa}(LG)$-module $\mathscr{C}$ an equivalence

$\mathscr{C}^{H,\phi}\simeq\operatorname{Hom}_{D_{\kappa}(LG)\operatorname{-mod}}(D_{\kappa}(LG)_{H,\phi},\mathscr{C}),$
(3.7)

where the appearing $(H,\phi)$-coinvariants are taken with respect to the
right $D(H)$ action induced by right multiplication.

######  Remark 3.8.2.

If we were to instead use the left $D(H)$ action induced by right
multiplication, we would rather consider $D_{\kappa}(LG)_{H,\phi^{\vee}}$,
where $\phi^{\vee}$ is the inverse character sheaf. For ease of notation, we
follow the former convention.

The equivalence (3.7) yields a $D_{\kappa}(LG)$-equivariant evaluation map

$D_{\kappa}(LG)_{H,\phi}\otimes\mathscr{C}^{H,\phi}\rightarrow\mathscr{C},\quad\quad
M\boxtimes c\mapsto M\overset{H}{\star}c.$

In particular, for a pair of such subgroups $(H_{i},\phi_{i},\tau_{i})$,
$i=1,2,$ one has a canonical map

$D_{\kappa}(LG)_{H_{2},\phi_{2}}^{H_{1},\phi_{1}}\otimes\mathscr{C}^{H_{2},\phi_{2}}\rightarrow\mathscr{C}^{H_{1},\phi_{1}}.$

If both $H_{1}$ and $H_{2}$ are as in Sections 3.7.3 and 3.7.4, so that we may
canonically identify their invariants and coinvariants, we write this as

$D_{\kappa}(H_{1},\phi_{1}\backslash
LG/H_{2},\phi_{2})\otimes\mathscr{C}^{H_{2},\phi_{2}}\rightarrow\mathscr{C}^{H_{1},\phi_{1}}.$

In the remainder of this section, we collect some basic notation for the
specific categories of bi-equivariant sheaves which will appear later.

###  3.9. Twisted affine Hecke categories

Fix a character sheaf $\chi\in\mathfrak{t}^{*}/\Lambda$, write $\mathring{I}$
for the prounipotent radical of $I$, and consider the category

$D_{\kappa}(\mathring{I}\backslash LG/I,\chi).$

Recall the Bruhat decomposition, i.e. that $LG$ admits a stratification with
strata $\mathring{I}wI,$ for $w\in W$. It is standard that the bi-equivariant
sheaves on a fixed stratum
$D_{\kappa}(\mathring{I}\backslash\mathring{I}wI/I,\chi)$ admit an equivalence
with $\operatorname{Vect}$. Indeed, for any coset representative $\dot{w}$ of
$w$, consider the projection and multiplication maps

$I\xleftarrow{\pi}\mathring{I}\times\dot{w}\times
I\xrightarrow{\mu}\mathring{I}wI.$

We may write $\pi^{!}\chi$ as $\mu^{!}\phi$, for some invertible sheaf $\phi$
on $\mathring{I}wI$. With this, the functor from $\operatorname{Vect}$ sending
$k$ to the twisted constant sheaf

$k(\mathring{I}wI)\overset{!}{\otimes}\phi$

is an equivalence. This is only canonical up to tensoring by a line in
cohomological degree zero, namely a $!$-fibre of $\chi$.

We denote the corresponding standard and costandard objects, i.e. the $!-$ and
$*$-extensions of $k[\ell(w)]$ in $\operatorname{Vect}$, by

$j_{w,!}^{w\chi,\chi}\text{ and }j_{w,*}^{w\chi,\chi}\in
D_{\kappa}(\mathring{I}\backslash LG/I,\chi),$

respectively. In particular, we will only define these objects up to tensoring
by a line in cohomological degree zero, which suffices for our purposes. We
also follow the practice of calling the objects $j_{w,*}^{w\chi,\chi}$
intertwining functors, in view of their function-theoretic counterparts.

Both $j_{w,!}^{w\chi,\chi}$ and $j_{w,*}^{w\chi,\chi}$ descend to
$D_{\kappa}(I,w\chi\backslash LG/I,\chi)$, where we use the level $\kappa$
action of $W$ on $\mathfrak{t}^{*}/\Lambda$ to obtain $w\chi$ from $\chi$.
Plainly, this action may be constructed by noting that the level $\kappa$ dot
action of $W$ on $\mathfrak{t}^{*}$, cf. Section 3.5.2, descends to a well-
defined action on $\mathfrak{t}^{*}/\Lambda.$

Fix any $\lambda\in\mathfrak{t}^{*}$ and real affine coroot $\check{\alpha}$
of $\widehat{\mathfrak{g}}_{\kappa}$. As $G$ is adjoint, and in particular
$\Lambda$ is the root lattice, it follows that

$s_{\check{\alpha}}\in W_{\lambda}\quad\text{if and only if}\quad
s_{\check{\alpha}}(\lambda+\Lambda)=\lambda+\Lambda,$

cf. Section 3.5.3 for a discussion of the integral Weyl group $W_{\lambda}$.
In particular, we may speak of the integral Weyl group $W_{\chi}$ of a twist
$\chi$. Namely, it is the subgroup of $W$ generated by the
$s_{\check{\alpha}}$ fixing $\chi$, or equivalently the integral Weyl group of
any lift $\tilde{\chi}\in\mathfrak{t}^{*}$.

###  3.10. Twisted Whittaker sheaves on the affine flag manifold

Finally, we would like to discuss some basic properties of the category

$D_{\kappa}(LN,\psi\backslash LG/I,\chi).$

Recall by the Iwasawa decomposition that $LG$ admits a stratification with
strata $LNwI,$ for $w\in W$. Denote by ${}^{f}W$ the subset of $W$ given by
elements of minimal length in their right $W_{f}$ cosets. The categories on
each stratum admit equivalences

$D_{\kappa}(LN,\psi\backslash
LNwI/I,\chi)\simeq\begin{cases}\operatorname{Vect}&\text{if $w\in{}^{f}W$},\\\
0&\text{otherwise}.\end{cases}$

To describe the former equivalences, again fix a coset representative
$\dot{w}$ and consider the multiplication

$LN\times I\simeq LN\times\dot{w}\times I\xrightarrow{\mu}LNwI.$

By the assumption that $w\in{}^{f}W$, we may write $\psi\boxtimes\chi$ as
$\mu^{!}\phi$, for an invertible sheaf $\phi$ on $LNwI$. The map from
$\operatorname{Vect}$ sending $k$ to the twisted constant sheaf

$k^{\psi,\chi}:=k(LNwI,L^{+}NwI)\overset{!}{\otimes}\phi$ (3.8)

is an equivalence, where the renormalized constant sheaf $k(LNwI,L^{+}NwI)$ is
defined as in Section 3.7.5. We denote the corresponding costandard object,
i.e. the $*$-extension of $k^{\psi,\chi}[\ell(w_{\circ}w)]$, by

$j_{w,*}^{\psi,\chi}.$ (3.9)

##  4\. Spectral flow and intertwining functors

The contents of this section are as follows. In the first half, we consider
the conjugations of Drinfeld–Sokolov reduction by arbitrary elements of $LG$.
When restricted to the category of highest weight modules, we find that the
resulting (nonzero) functors are in canonical bijection with the double cosets
in

$LN\backslash LG/I$

which support Whittaker sheaves. Concretely, these are parametrized by the
coweight lattice, and the corresponding functors are essentially the usual
spectrally flowed reduction functors, see Remark 4.4.3 below for the precise
relation.

In the second half, we provide a canonical factorization of each spectrally
flowed reduction applied to a block of highest weight modules. These depend on
the block and refine the fact, which essentially appears in Frenkel–Gaitsgory
[FG06], that the various reductions all differ from one another by
intertwining functors. As we will see in Section 5, these factorizations
underlie the combinatorics of the Frenkel–Kac–Wakimoto conjecture for the plus
reduction.

### 4.1.

For a generic character $\psi$ of $LN$ of conductor one, cf. Section 3.4.2,
consider the semi-infinite cohomology functor

$\mathring{\Psi}^{e}:=C^{\frac{\infty}{2}+*}(L\mathfrak{n},L^{+}\mathfrak{n},-\otimes
k_{\psi}):\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\rightarrow\mathcal{W}_{\kappa}\operatorname{-mod}.$

With our intended applications to the plus reduction in mind, we will consider
in this section the conjugates of $\mathring{\Psi}^{e}$ by elements of $LG$,
and study their restriction to highest weight modules.

######  Remark 4.1.1.

As the minus reduction $\Psi^{-}$ is normalized instead with respect to the
first congruence subalgebra of $L^{+}\mathfrak{n}$, by trivializing its
relative determinant with $L^{+}\mathfrak{n}$, cf. Section 3.2.1, one obtains
an equivalence

$\mathring{\Psi}^{e}[\ell(w_{\circ})]\simeq\Psi^{-}.$

### 4.2.

Fix any element $g\in LG$. Conjugation by $g$ yields an automorphism
$\operatorname{Ad}_{g}:\widehat{\mathfrak{g}}_{\kappa}\rightarrow\widehat{\mathfrak{g}}_{\kappa}.$
We denote the corresponding pushforward of modules, i.e. restriction along
$\operatorname{Ad}_{g^{-1}}$, by

$\operatorname{Ad}_{g,*}:\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\rightarrow\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}.$

We denote by $\mathring{\Psi}^{g}$ its composition with the previous semi-
infinite cohomology functor, i.e.

$\mathring{\Psi}^{g}:\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\xrightarrow{\operatorname{Ad}_{g,*}}\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\xrightarrow{\mathring{\Psi}^{e}}\mathcal{W}_{\kappa}\operatorname{-mod}.$
(4.1)

######  Example 4.2.1.

Fix a coset representative $\dot{w}_{\circ}$ in $N_{G}(T)$ for the longest
element $w_{\circ}$ of $W_{f}$. Let us specialize the preceding discussion to
$g=\dot{w}_{\circ}t^{\check{\rho}}$. In this case (4.1) is, up to a
cohomological shift coming from the normalizing compact open subalgebras, the
plus Drinfeld–Sokolov reduction.

More precisely, as the plus reduction is normalized with respect to
$L^{+}\mathfrak{n}^{-}$, rather than

$\operatorname{Ad}_{t^{-\check{\rho}}}L^{+}\mathfrak{n}^{-},$

by trivializing the appearing relative determinant line we obtain an
isomorphism

$\mathring{\Psi}^{\dot{w}_{\circ}t^{\check{\rho}}}[2\langle\rho,\check{\rho}\rangle]\simeq\Psi^{+}.\
$

### 4.3.

We will be interested in the behavior of the functor (4.1) on highest weight
modules. More precisely, let us fix $\chi\in\mathfrak{t}^{*}/\Lambda$, which
we identify with the corresponding character sheaf on $I$, and consider the
functor

$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}\xrightarrow{\operatorname{Oblv}}\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\xrightarrow{\operatorname{Ad}_{g,*}}\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}\xrightarrow{\mathring{\Psi}^{e}}\mathcal{W}_{\kappa}\operatorname{-mod},$
(4.2)

cf. Section 3.5.4 for a discussion of
$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}$. We will rewrite
(4.2) as follows. For any $D_{\kappa}(LG)$-module module $\mathscr{C}$, we may
form the composition

$\mathscr{C}^{I,\chi}\xrightarrow{\operatorname{Oblv}}\mathscr{C}\xrightarrow{\delta_{g}\star-}\mathscr{C}\xrightarrow{\operatorname{Av}_{*}^{\psi,r}}\mathscr{C}^{LN,\psi};$
(4.3)

the reader may consult Section 3.6 for a discussion of the appearing terms.
Taking $\mathscr{C}=\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}$ then
recovers (4.2). In particular, (4.2) is given by convolution with an object

$\mathring{\mathscr{K}}^{g}\in D_{\kappa}(LN,\psi\backslash LG/I,\chi).$

To identify it explicitly, recall that on a stratum $LNgI$ admitting nonzero
Whittaker sheaves we denote by $k^{\psi,\chi}$ the $(\psi,\chi)$-twist of its
renormalized constant sheaf, cf. Equation (3.8).

######  Proposition 4.3.1.

Consider the double coset $LNgI$. If this does not support Whittaker sheaves,
i.e.

$D(LN,\psi\backslash LNgI/I,\chi)\simeq 0,$

then $\mathring{\mathscr{K}}^{g}$ vanishes. Otherwise,
$\mathring{\mathscr{K}}^{g}$ is the $*$-extension of the renormalized constant
sheaf $k^{\psi,\chi}$ on $LNgI$.

###### Proof.

It follows from (4.3) that $\mathring{\mathscr{K}}^{g}$ is the $*$-pushforward
along the multiplication map

$LN\times I\simeq LN\times\\{g\\}\times I\xrightarrow{\mu}LNgI\hookrightarrow
LG$

of the sheaf $k^{\psi,r}\boxtimes\chi$, where $k^{\psi,r}$ is the kernel for
the renormalized Whittaker averaging functor $\operatorname{Av}_{*}^{\psi,r}$,
cf. Sections 3.7.4 and 3.7.5 for further discussion. As $\mu$ is a fibration
by prounipotent groups and is compatible with the trivializations of dimension
torsors used in forming $k^{\psi,r}$ and $k^{\psi,\chi}$, cf. Equations (3.6)
and (3.8) respectively, the lemma follows. ∎

### 4.4.

By Proposition 4.3.1, the functor (4.2) depends only on the double coset
$LNgI$. In particular, if $\mathring{\mathscr{K}}^{g}$ is nonvanishing, we may
write

$LNgI=LNwI\quad\quad\text{for some }w\in{}^{f}W,$ (4.4)

where as in Section 3.10 we denote by ${}^{f}W$ the elements of the extended
affine Weyl group of minimal length in their right $W_{f}$ cosets.

Therefore, if we introduce the cohomological shifts
$\Psi^{g}:=\mathring{\Psi}^{g}[\ell(w_{\circ}w)]$ and
$\mathscr{K}_{g}:=\mathring{\mathscr{K}}^{g}[\ell(w_{\circ}w)]$, we may deduce
the following from Proposition 4.3.1.

######  Corollary 4.4.1.

For $g$ and $w$ as in Equation (4.4), there exists an isomorphism

$\mathscr{K}^{g}\simeq j_{{w},*}^{\psi,\chi},$ (4.5)

where $j_{w,*}^{\psi,\chi}$ is as in Equation (3.9).

In what follows, we refer to such functors as the spectrally flowed
Drinfeld–Sokolov reductions.

######  Example 4.4.2.

As important special cases, Corollary 4.4.1 identifies the plus and minus
reductions with the corresponding costandard objects. Namely, for the minus
reduction, it follows from Remark 4.1.1 and Corollary 4.4.1 that one has
isomorphisms

$\Psi^{-}\simeq\Psi^{e}\quad\text{and}\quad\mathscr{K}^{e}\simeq
j_{e,*}^{\psi,\chi}.$

For the plus reduction, note that the element $w_{\circ}e^{\check{\rho}}\in W$
lies in ${}^{f}W$, and has length

$2\langle\rho,\check{\rho}\rangle-\ell(w_{\circ}).$

It therefore follows from Example 4.2.1 and Corollary 4.4.1 that one has
isomorphisms

$\Psi^{+}\simeq\Psi^{\dot{w}_{\circ}t^{\check{\rho}}}\quad\text{and}\quad\mathscr{K}^{\dot{w}_{\circ}t^{\check{\rho}}}\simeq
j_{w_{\circ}e^{\check{\rho}},*}^{\psi,\chi}.$

######  Remark 4.4.3.

We should mention that our usage of the term spectrally flowed reduction is
slightly nonstandard. To explain this, consider the tautological
identifications

${}^{f}W\hskip 2.84544pt\simeq\hskip 2.84544ptW_{f}\backslash W\hskip
2.84544pt\simeq\hskip 2.84544pt\check{\Lambda}.$

For any $\check{\mu}\in\check{\Lambda}$ let us write $we^{\check{\mu}}$ for
the corresponding element of ${}^{f}W$, i.e. the minimal length element in its
right $W_{f}$ coset. By definition, our associated reduction functor

$\Psi^{\check{\mu}}:=\Psi^{\dot{w}t^{\check{\mu}}}$

is given up to an overall cohomological shift by (i) first applying the
spectral flow $\operatorname{Ad}_{t^{\check{\mu}}}$ to a
$\widehat{\mathfrak{g}}_{\kappa}$-module, (ii) further conjugating by
$\operatorname{Ad}_{\dot{w}_{f}}$, and then (iii) applying the minus
reduction. We note that if $\check{\mu}$ is antidominant, step (ii) is
trivial, in which case this agrees up to notational conventions with the
standard usage, see e.g. [AF19].

### 4.5.

In the remainder of this section we would like to record a canonical
factorization, for any $w\in{}^{f}W$, of the corresponding costandard object

$j_{{w},*}^{\psi,\chi}.$

We begin with the factorization at the level of the affine Weyl group.
Consider the integral Weyl group $W_{\chi}$ of the character sheaf $\chi$, cf.
the final paragraphs of Section 3.9. Recall this is a Coxeter system, i.e. a
Coxeter group with a preferred set of generators.

######  Lemma 4.5.1.

For any $w\in W$, there is a unique element $w_{-}$ of minimal length in the
double coset

$W_{f}wW_{\chi}.$ (4.6)

Moreover, the $w_{-}$-conjugated intersection

$w_{-}^{-1}W_{f}w_{-}\cap W_{\chi}$ (4.7)

is a parabolic subgroup of $W_{\chi}$. In particular, if the subgroup (4.7) is
trivial, $w$ admits a unique factorization

$w=w_{f}w_{-}w_{\chi},\quad\quad\quad\text{where $w_{f}\in W_{f}$ and
$w_{\chi}\in W_{\chi}$.}$ (4.8)

######  Remark 4.5.2.

To orient the reader, the relation of the statement of the lemma to
Frenkel–Kac–Wakimoto’s conjecture will be as follows. The group appearing in
Equation (4.7) will be nontrivial if and only if the reduction vanishes. When
it is trivial, the canonical factorization of Equation (4.8), and in
particular the length of $w_{\chi}$ in the Coxeter system $W_{\chi}$, will
produce the cohomological shift of the conjecture.

###### Proof of Lemma 4.8.

We proved the first two assertions in Lemma 3.2.9 of [CDRar]. The final
statement follows by noting that the stabilizer of $w_{-}$ under the right
action

$W\times W_{f}\times W_{\chi}\rightarrow W,\quad\quad
y\cdot(w_{f},w_{\chi})=w_{f}^{-1}yw_{\chi},$

is identified via the projection onto $W_{\chi}$ with (4.7). . ∎

We next lift Lemma 4.8 to a statement about costandard objects as follows. Fix
an element $w\in{}^{f}W$, and consider the projection

$\pi:W\rightarrow W_{f}\backslash W.$

By Lemma 4.8, we have a canonical identification of the arising
$W_{\chi}$-orbit as

$\pi(w)\cdot W_{\chi}\simeq(w_{-}^{-1}W_{f}w_{-}\cap W_{\chi})\backslash
W_{\chi},$

which is afforded by acting on the minimal length element $\pi(w_{-})$. As the
stabilizer of the latter is a parabolic subgroup of $W_{\chi}$, we may
canonically write

$\pi(w)=\pi(w_{-})\cdot w_{\chi},$ (4.9)

where $w_{\chi}\in W_{\chi}$ is the unique element of minimal length in its
right $w_{-}^{-1}W_{f}w_{-}\cap W_{\chi}$ coset. With these preparations, we
may describe the lift to intertwining functors.

######  Proposition 4.5.3.

Let $w$, $w_{-},w_{\chi}$ be as in the preceding paragraph. There is an
isomorphism

$j_{{w},*}^{\psi,\chi}\hskip 2.84526pt\simeq\hskip
2.84526ptj_{e,*}^{\psi,w_{-}\chi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptj_{{w}_{-},*}^{w_{-}\chi,\chi}\hskip
2.84526pt\overset{I}{\star}\hskip 2.84526ptj_{{w}_{\chi},*}^{\chi,\chi},$

and the appearing object $j_{{w}_{-},*}^{w_{-}\chi,\chi}$ is clean, i.e.
coincides with the standard object $j_{{w}_{-},!}^{w_{-}\chi,\chi}$.

###### Proof.

We will obtain the desired factorization in two steps. First, we exhibited an
isomorphism

$j_{{w},*}^{\psi,\chi}\hskip 2.84526pt\simeq\hskip
2.84526ptj_{{w}_{-},*}^{\psi,\chi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptj_{{w}_{\chi},*}^{\chi,\chi}$ (4.10)

in Proposition 3.2.12 of [CDRar].777To aid the reader consulting loc. cit., we
note that the subgroups $N$ and $N^{-}$ of the present paper are denoted there
by $N^{-}$ and $N$, respectively. Second, we may further factor the right-hand
side via an isomorphism

$j_{{w}_{-},*}^{\psi,\chi}\hskip 2.84526pt\simeq\hskip
2.84526ptj_{e,*}^{\psi,w_{-}\chi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptj_{{w}_{-},*}^{w_{-}\chi,\chi},$

which along with the cleanness of $j_{{w}_{-},*}^{w_{-}\chi,\chi}$ we recorded
in loc. cit. as Proposition 3.3.8. ∎

### 4.6.

Explicitly, if we combine Remark 4.1.1, Corollary 4.4.1 and Proposition 4.5.3,
we obtain the following.

######  Corollary 4.6.1.

Fix an element $w\in{}^{f}W$ along with a coset representative $\dot{w}$.
Then, for any character $\chi$, the spectrally flowed Drinfeld–Sokolov
reduction

$\Psi^{\dot{w}}:\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}\rightarrow\mathcal{W}_{\kappa}\operatorname{-mod}$

is naturally isomorphic to the composition

$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}\xrightarrow{j_{{w}_{\chi},*}^{\chi,\chi}}\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}\xrightarrow{j_{{w}_{-},*}^{w_{-}\chi,\chi}}\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,w_{-}\chi}\xrightarrow{\Psi^{-}}\mathcal{W}_{\kappa}\operatorname{-mod},$

where $w_{\chi}$ and $w_{-}$ are as in Proposition 4.5.3. That is, the
spectrally flowed reduction is a composition of intertwining functors and the
minus reduction.

In particular, taking $w=w_{\circ}e^{\check{\rho}}$, the above rewrites the
plus reduction via intertwining functors and the minus reduction, cf. Example
4.4.2. Let us now apply this to our motivating problem.

##  5\. The Frenkel–Kac–Wakimoto conjecture

In this section, we calculate the spectrally flowed reductions of certain
irreducible highest weight modules. The structure is as follows. Bearing in
mind the factorization of Corollary 4.6.1, we first establish some preliminary
assertions about the behavior of the appearing intertwining functors. With
these in hand, the main result is obtained in Theorem 5.6.1.

### 5.1.

Let us begin by establishing notation. Suppose that
$\lambda\in\mathfrak{t}^{*}$ is a dominant weight at level $\kappa$. That is,
we assume the Verma module $M_{\lambda}$ is projective in the Category
$\mathscr{O}$ of $\widehat{\mathfrak{g}}_{\kappa}$-modules. By the theorem of
Kac–Kazhdan on singular vectors [KK79], this is equivalent to the condition
that $\kappa$ is noncritical and for every positive real affine coroot
$\check{\alpha}$ one has

$\langle\hskip 0.85358pt\check{\alpha},\lambda\rangle\notin\mathbb{Z}^{<0},$

cf. Section 3.5.3 for a discussion of the appearing pairing.

We would like to calculate the spectrally flowed Drinfeld–Sokolov reductions
of the corresponding irreducible module

$\Psi^{\dot{w}}(L_{\lambda}),\quad\quad\text{for }w\in{}^{f}W,$

where $\Psi^{\dot{w}}$ is as in Section 4.4.

### 5.2.

To do so, write $\chi$ for the character sheaf on $I$ associated to the root
lattice coset $\lambda+\Lambda\in\mathfrak{t}^{*}/\Lambda$, so that

$L_{\lambda}\in\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}.$

Recall from Corollary 4.6.1 the associated factorization

$\Psi^{\dot{w}}\simeq\Psi^{-}\circ j_{{w}_{-},*}^{w_{-}\chi,\chi}\circ
j_{{w}_{\chi},*}^{\chi,\chi}.$ (5.1)

We now collect some facts about the behavior of the first steps, i.e. the
intertwining functors.

### 5.3.

It will be convenient to begin with the middle factor
$j_{{w}_{-},*}^{w_{-}\chi,\chi}$. Recalling it is clean, we will control it as
follows.

######  Proposition 5.3.1.

Fix a character sheaf $\phi\in\mathfrak{t}^{*}/\Lambda$ and an element $w\in
W$. Suppose that the associated intertwining integral $j_{w,*}^{w\phi,\phi}$
is clean. Then the corresponding convolution functor

$j_{w,*}^{w\phi,\phi}:\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\phi}\rightarrow\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,w\phi}$

is $t$-exact, and for any $\mu\in\phi+\Lambda$ there are isomorphisms

$j_{w,*}^{w\phi,\phi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptM_{\mu}\simeq M_{w\mu}\quad\text{and}\quad j_{w,*}^{w\phi,\phi}\hskip
2.84526pt\overset{I}{\star}\hskip 2.84526ptL_{\mu}\simeq L_{w\mu}.$ (5.2)

######  Remark 5.3.2.

It formally follows that the convolution also exchanges the contragredient
Verma modules.

###### Proof of Proposition 5.3.1.

We first reduce the assertion to the case where $w$ is a simple reflection. If
$w$ has length zero, the assertion is clear. If $w$ has length greater than
one, then $w>ws$ for some simple reflection $s$, and we have a corresponding
factorization

$j_{w,*}^{w\phi,\phi}\hskip 2.84526pt\simeq\hskip
2.84526ptj_{ws,*}^{w\phi,s\phi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptj_{s,*}^{s\phi,\phi}.$ (5.3)

We claim that both factors of the right-hand side are again clean. We first
show this for $j_{s,*}^{s\phi,\phi}$. Indeed, if not, we would have
$s\phi=\phi$ and a surjection $j_{s,*}^{\phi,\phi}\twoheadrightarrow
j_{e,*}^{\phi,\phi}$. By convolving this with $j_{w,*}^{w\phi,\phi}$, by the
right exactness of convolution with costandard objects we would obtain a
surjection

$j_{w,*}^{w\phi,\phi}\twoheadrightarrow j_{ws,*}^{w\phi,\phi},$

which would contradict the cleanness of $j_{w,*}^{w\phi,\phi}$.

To proceed, we will use the following basic lemma.

######  Lemma 5.3.3.

Suppose $s$ is a simple reflection of $W,$ and $\phi$ is a character sheaf
such that the object $j_{s,*}^{s\phi,\phi}$ is clean. Then for any $w\in W$
there are isomorphisms

$j_{w,!}^{w\phi,s\phi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptj_{s,!}^{s\phi,\phi}\hskip 2.84526pt\simeq\hskip
2.84526ptj_{ws,!}^{w\phi,\phi}\quad\text{and}\quad j_{w,*}^{w\phi,s\phi}\hskip
2.84526pt\overset{I}{\star}\hskip 2.84526ptj_{s,*}^{s\phi,\phi}\hskip
2.84526pt\simeq\hskip 2.84526ptj_{ws,*}^{w\phi,\phi}$ (5.4)

That is, convolution with $j_{s,*}^{s\phi,\phi}$ sends (co)standard objects to
(co)standard objects.

######  Remark 5.3.4.

This lemma appears, mutatis mutandis, in the study of finite twisted Hecke
categories performed in [LY20], and likely earlier as well. A similar analysis
of the affine twisted Hecke categories, which refines certain results proved
in the remainder of the present paper, as well as its applications, e.g. to
the tamely ramified local quantum Langlands correspondence, will appear in
forthcoming work of the author and Y. W. Li, Z. Yun, and X. Zhu.

###### Proof of Lemma 5.3.3.

If $w<ws$, then the both assertions of Equation (5.4) are well known to hold
without assumptions of cleaness, and follow from the fact that

$\mathring{I}wI\overset{I}{\times}IsI\rightarrow\mathring{I}wsI$

is an isomorphism. The claims for $w>ws$ then follow from the fact that
$-\hskip 2.84526pt\overset{I}{\star}\hskip 2.84526ptj_{s,*}^{s\phi,\phi}$ is
inverse to $-\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptj_{s,!}^{\phi,s\phi}$, and the cleanness of the latter. ∎

With the lemma in hand, the asserted cleanness of $j_{ws,*}^{w\phi,s\phi}$
follows from (5.3).

Iterating the factorization (5.3), it follows that $j_{w,*}^{w\phi,\phi}$ may
be written as a convolution of clean costandard objects associated to simple
reflections and a length zero costandard object. As the claims of the
proposition are compatible with composition, this completes the reduction to
the case of $w$ being a simple reflection $s$.

In this case, write $P$ for the corresponding (minimal) standard parahoric
subgroup of $LG$. We first argue that for any $\mu\in\phi+\Lambda$ one has

$j_{s,*}^{s\phi,\phi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptM_{\mu}\simeq M_{s\mu}.$ (5.5)

To see this, write $M$ and $\mathfrak{m}$ for the corresponding Levi factors
of $LG$ and $\widehat{\mathfrak{g}}_{\kappa}$, respectively, and note these
are of semisimple rank one. By the $D(P)$-equivariance of parabolic induction

$\operatorname{pind}:\mathfrak{m}\operatorname{-mod}\rightarrow\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod},$

see for example Section 2.5.4 of [CD21], the assertion (5.5) reduces to the
analogous assertion for $\mathfrak{m}$. To see the latter, write $B^{s}$ for
the opposite Borel of $P$ containing $T$, and $N^{s}$ for its prounipotent
radical. If we pick any lift $\dot{s}$ of $s$ to $N_{M}(T)$, we may identify
$j_{s,*}^{s\phi,\phi}$ with the composition

$\mathfrak{m}\operatorname{-mod}^{B,\phi}\xrightarrow{\operatorname{Av}_{N^{s},*}[1]}\mathfrak{m}\operatorname{-mod}^{B^{s},s\phi}\xrightarrow{\delta_{\dot{s}}}\mathfrak{m}\operatorname{-mod}^{B,s\phi}.$
(5.6)

The remainder follows from the highest weight representation theory of
$\mathfrak{sl}_{2}$. Briefly, by our assumptions on $\phi$ and $s\phi$, the
appearing categories are semisimple, so one is reduced to a straightforward
verification at the level of characters.

It remains to prove the $t$-exactness of $j_{s,*}^{s\phi,\phi}$ and the
interchanging of simple modules. But recall that an object $N$ of
$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,s\phi}$ lies in
cohomological degrees $\geqslant 0$ if and only if

$\operatorname{Hom}(M_{\nu},N)\xrightarrow{\sim}\tau^{\geqslant
0}\operatorname{Hom}(M_{\nu},N)$

$\text{for all $\nu$ in $s\phi+\Lambda$}.$ Therefore, the claimed
$t$-exactness of $j_{s,*}^{s\phi,\phi}$ follows from (5.5). Moreover, as
$j_{s,*}^{s\phi,\phi}$ is further an equivalence, it also exchanges the simple
quotients of the Verma modules, as desired. ∎

### 5.4.

We next gather some facts which will be of use in controlling the final
intertwining integral on the right-hand side of (5.1), namely
$j_{w_{\chi},*}^{\chi,\chi}$.

Recall that $W_{\chi}$ is a Coxeter group, and in particular carries a
distinguished set of simple reflections and a Bruhat order. We will need the
following assertion.

######  Proposition 5.4.1.

Fix elements $y,s\in W_{\chi}$ such that $s$ is a simple reflection of
$W_{\chi}$ and $y<ys$. There exists an isomorphism of costandard objects

$j_{ys,*}^{\chi,\chi}\hskip 2.84526pt\simeq\hskip
2.84526ptj_{y,*}^{\chi,\chi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptj_{s,*}^{\chi,\chi}.$

We emphasize that the appearing simple reflection $s$ of $W_{\chi}$ typically
is not a simple reflection of the ambient group $W$. This renders the
proposition not entirely geometrically obvious, as the underlying convolution
of Schubert varieties will typically not be an isomorphism. Before giving the
proof, we note the following useful consequence.

######  Corollary 5.4.2.

For an element $y$ of $W_{\chi}$, fix a reduced expression

$y=s_{i_{1}}s_{i_{2}}\cdots s_{i_{\ell}}.$

There exists a corresponding isomorphism of costandard objects

$j_{\dot{y},*}^{\chi,\chi}\hskip 2.84526pt\simeq\hskip
2.84526ptj_{s_{i_{1}},*}^{\chi,\chi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptj_{s_{i_{2}},*}^{\chi,\chi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526pt\cdots\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptj_{s_{i_{\ell}},*}^{\chi,\chi}.$

###### Proof of Proposition 5.4.1.

We will reduce to the case of $s$ being also a simple reflection of $W$ using
the following lemma.

######  Lemma 5.4.3.

If $s$ is a simple reflection of $W_{\chi}$, then there exists elements $z$
and $t$ in $W$ such that the following hold.

  1. (1)

The element $t$ is a simple reflection of $W$.

  2. (2)

One has the equality $s=ztz^{-1}$.

  3. (3)

The action of $z$ on the real affine coroots restricts to an isomorphism

$z:\check{\Phi}_{\chi}^{+}\xrightarrow{\sim}\check{\Phi}_{z\chi}^{+}.$ (5.7)

In particular, conjugation by $z$ yields an isomorphism of Coxeter systems

$z:W_{\chi}\xrightarrow{\sim}W_{z\chi}.$

###### Proof.

If the coroot $\check{\alpha}$ corresponding to $s$ is simple in
$\check{\Phi}^{+}$ we are done. If not, there exists a simple reflection $u$
of $W$ such that

$\check{\alpha}>u(\check{\alpha})>0.$ (5.8)

Since by assumption $\check{\alpha}$ is a simple coroot in
$\check{\Phi}^{+}_{\chi}$, it follows that $u$ does not lie in $W_{\chi}$, and
in particular we have that acting by $u$ restricts to an isomorphism

$u:\check{\Phi}^{+}_{\chi}\xrightarrow{\sim}\check{\Phi}^{+}_{u\chi}.$

This reduces the assertions of the lemma for the pair $(\chi,s)$ to those for
$(u\chi,usu^{-1})$. By Equation (5.8) and the fact that there are only
finitely many positive coroots less than $\check{\alpha}$, iterating this
argument must bring us to a simple coroot of $\check{\Phi}^{+}$, as desired. ∎

Let $s$ be as in the statement of the proposition, and let $t$ and $z$ be as
in the statement of Lemma 5.4.3. By the proof of the lemma, in combination
with Lemma 5.3.3, it follows that the objects

$j_{z,*}^{\chi,z\chi}\quad\text{and}\quad j_{z^{-1},*}^{z\chi,z}$

are clean. In particular, the conjugation monoidal equivalence

$j_{z,*}^{\chi,z\chi}\hskip 2.84526pt\overset{I}{\star}\hskip 2.84526pt-\hskip
2.84526pt\overset{I}{\star}\hskip
2.84526ptj_{z^{-1},*}^{z\chi,\chi}:D(I,z\chi\backslash LG/I,z\chi)\simeq
D(I,\chi\backslash LG/I,\chi)$

reduces the assertion of the proposition to the case where the simple
reflection $s$ of $W_{\chi}$ is a simple reflection of $W$, which is standard,
cf. Lemma 5.3.3. ∎

Since we will use it later, we explicitly record one fact developed in the
proof of Proposition 5.4.1. Namely, the following lifts Lemma 5.4.3,
particularly statement (2) therein, to the intertwining functors.

######  Lemma 5.4.4.

Let $s,t,z$ be as in Lemma 5.4.3. Then there is an isomorphism of costandard
objects

$j_{s,*}^{\chi,\chi}\hskip 2.84526pt\simeq\hskip
2.84526ptj_{z,*}^{\chi,z\chi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptj_{t,*}^{z\chi,z\chi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptj_{z^{-1},*}^{z\chi,\chi}.$

That is, up to conjugation by a clean object, any costandard object associated
to a simple reflection of $W_{\chi}$ is a costandard object associated to a
simple reflection of $W$.

### 5.5.

In this subsection, which may be skipped by the reader, we use what we have
developed so far to obtain bounds on the cohomological amplitude of the
spectrally flowed reduction functors. To our knowledge, these are new already
for the plus reduction.

Let us collect two pieces of notation. First, we denote the standard length
function on $W_{\chi}$ with respect to its Coxeter generators by

$\ell_{\chi}:W_{\chi}\rightarrow\mathbb{Z}^{\geqslant 0}.$

Second, let $\mathscr{C}$ and $\mathscr{D}$ be dg-categories equipped with
$t$-structures and a functor

$F:\mathscr{C}\rightarrow\mathscr{D}.$

Fix integers $a$ and $b$ with $a\leqslant b$. Recall that $F$ is said to have
cohomological amplitude at most $[a,b]$ if $F[b]$ is right exact and $F[a]$ is
left exact, i.e.

$F(\mathscr{C}^{\leqslant 0})\subset\mathscr{D}^{\leqslant
b}\quad\text{and}\quad F(\mathscr{C}^{\geqslant
0})\subset\mathscr{D}^{\geqslant a}.$

In particular, for any object $\xi$ in the heart of the $t$-structure on
$\mathscr{C}$, $F(\xi)$ has nonzero cohomology, with respect to the truncation
functors on $\mathscr{D}$, in at most cohomological degrees $a\leqslant
i\leqslant b$.

######  Theorem 5.5.1.

Let $w,\chi,w_{\chi}$ be as in Proposition 4.5.3. Then the spectrally flowed
reduction functor

$\Psi^{w}:\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}\rightarrow\mathcal{W}_{\kappa}\operatorname{-mod}$

has cohomological amplitude at most

$[-\ell_{\chi}(w_{\chi}),0].$ (5.9)

######  Remark 5.5.2.

The bounds (5.9) are essentially optimal. On the one hand, the upper bound is
always obtained; indeed it follows from Corollary 4.6.1 that $\Psi^{w}$ sends
Verma modules to twisted Verma modules concentrated in degree zero. On the
other hand, we will see momentarily that reductions of dominant irreducibles,
when nonzero, are concentrated in degree $-\ell_{\chi}(w_{\chi})$.

###### Proof of Theorem 5.5.1.

In view of Corollary 4.6.1, we equivalently must show that the composition

$\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}\xrightarrow{j_{{w}_{\chi},*}^{\chi,\chi}}\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}\xrightarrow{j_{{w}_{-},*}^{w_{-}\chi,\chi}}\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,w_{-}\chi}\xrightarrow{\Psi^{-}}\mathcal{W}_{\kappa}\operatorname{-mod},$

has cohomological amplitude at most $[-\ell_{\chi}(w_{\chi}),0]$. Recall that
the middle and rightmost functors are $t$-exact, see Proposition 5.3.1 and
Theorem 3.5.6, respectively. Therefore, it is enough to show that
$j_{{w}_{\chi},*}^{\chi,\chi}$ has cohomological amplitude at most

$[-\ell_{\chi}(w_{\chi}),0].$

To see this, by Corollary 5.4.2 it suffices to show that for a simple
reflection $s$ of $W_{\chi}$ the intertwining integral $j_{s,*}^{\chi,\chi}$
has cohomological amplitude at most

$[-1,0].$

By Lemma 5.4.4 and Proposition 5.3.1, we may assume that $s$ is also a simple
reflection of $W$. However, this final case is clear, for example by the
analog of Equation (5.6). ∎

### 5.6.

To state our main theorem, we now introduce one final piece of notation, and
then review several others.

Let us write $D_{\kappa}$ for the set of dominant weights at level $\kappa$.
This carries an action of $W$, which we denote by

$W\times D_{\kappa}\rightarrow D_{\kappa},\quad\quad(w,\lambda)\mapsto
w*\lambda,$

defined as follows. Given a pair $(w,\lambda)$, the orbit of $w\lambda$ under
its integral Weyl group contains a unique dominant weight $w\lambda^{+}$, and
we set

$w*\lambda:=w\lambda^{+}.$

It is straightforward that this defines an action.

For the statement of the following theorem we recall that $\pi$ denotes the
Harish-Chandra projection

$\pi:\mathfrak{t}^{*}\rightarrow\mathfrak{t}^{*}//W_{f},$

and that a weight $\mu\in\mathfrak{t}^{*}$ is said to be finite antidominant
if for every finite positive coroot $\check{\alpha}$ one has

$\langle\check{\alpha},\lambda\rangle\notin\mathbb{Z}^{>0},$

where the appearing pairing incorporates a $\rho$-shift as in Section 3.5.3.
Finally, we refer to Section 2.2 for our normalization of highest weights for
$\mathcal{W}_{\kappa}$-modules. We are now ready to prove our main result.

######  Theorem 5.6.1.

Fix an element $w\in{}^{f}W$ and a dominant irreducible module

$L_{\lambda}\in\widehat{\mathfrak{g}}_{\kappa}\operatorname{-mod}^{I,\chi}.$

Let $w_{\chi}$ be as in Proposition 4.5.3. Then the corresponding spectrally
flowed Drinfeld–Sokolov reduction is given by

$\Psi^{\dot{w}}(L_{\lambda})\simeq\begin{cases}L_{\pi(w*\lambda)}[\ell_{\chi}(w_{\chi})]&\text{if
$w*\lambda$ is finite antidominant,}\\\ 0&\text{otherwise}.\end{cases}$

###### Proof.

We will proceed step by step along the factorization of Corollary 4.6.1,
namely

$\Psi^{w}\simeq\Psi^{-}\circ j_{w_{-},*}^{w_{-}\chi,\chi}\circ
j_{w_{\chi},*}^{\chi,\chi}.$

To begin, we claim that one has an isomorphism

$j_{w_{\chi},*}^{\chi,\chi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptL_{\lambda}\simeq L_{\lambda}[\ell_{\chi}(w_{\chi})].$ (5.10)

To see this, by Corollary 5.4.2 it is enough to show that for a simple
reflection $s$ of $W_{\chi}$ one has

$j_{s,*}^{\chi,\chi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptL_{\lambda}\simeq L_{\lambda}[1].$

By Lemma 5.4.4 and Proposition 5.3.1, we may further assume that $s$ is a
simple reflection of $W$. In this case, let $P$ be the corresponding parahoric
subgroup of $LG$, and note that $\chi$ extends uniquely to a character sheaf
on $P$, which we denote by the same letter.

As $L_{\lambda}$ is $(P,\chi)$-equivariant, it suffices to see that for any
$D_{\kappa}(LG)$-module $\mathscr{C}$ the composition

$\mathscr{C}^{P,\chi}\xrightarrow{\operatorname{Oblv}}\mathscr{C}^{I,\chi}\xrightarrow{j_{s,*}^{\chi,\chi}}\mathscr{C}^{I,\chi}$
(5.11)

is equivalent to

$\mathscr{C}^{P,\chi}\xrightarrow{\operatorname{Oblv}[1]}\mathscr{C}^{I,\chi}.$
(5.12)

However, it is straightforward to see by a calculation on the Levi factor of
$P$, which is of semisimple rank one, that both (5.11) and (5.12) are given by
convolution with the same object of

$D_{\kappa}(I,\chi\backslash LG/P,\chi),$

as desired. This completes the proof of (5.10).

To proceed, recall that $j_{w_{-},*}^{w_{-}\chi,\chi}$ is clean. By
Proposition 5.3.1, it therefore follows that

$j_{w_{-},*}^{w_{-},\chi}\hskip 2.84526pt\overset{I}{\star}\hskip
2.84526ptL_{\lambda}[\ell_{\chi}(w_{\chi})]\simeq
L_{w_{-}\lambda}[\ell_{\chi}(w_{\chi})].$

Finally, applying the minus reduction $\Psi^{-}$ to
$L_{w_{-}\lambda}[\ell_{\chi}(w_{\chi})]$ yields the statement of the theorem.
∎

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