Research Statement Sam Gunningham

My research is centered around the field of Geometric Representation Theory. Much of my work involves the study of algebraic systems of differential equations (D-modules) in the presence of symmetry, drawing on concepts and tools from category theory, homotopy theory, derived algebraic geometry, and mathematical physics. The first section of this statement gives an informal overview of my research, with a more detailed account following in the subsequent sections. 1 Overview: Categorical Harmonic Analysis Briefly, the goal of harmonic analysis is to express functions as superpositions of “basic waves”. In categorical harmonic analysis, the functions are replaced by differential equations (or alternatively their sheaves of solutions). This is motivated by the observation that the “basic waves” are typically solutions of “basic eigenproblems” (for example, the exponential function upxq “ eiξx is an eigenfunction for the polynomial differential operator B{Bx). The main object of study is typically the category DpXq of D-modules (algebraic incarnations of differential equations) on a space X. The theory of reductive groups has long been a fertile setting for harmonic analysis (after pioneering work of Harish-Chandra, Langlands, and others) and much of my own work is set in this world. For example, many of my projects are centered around the category DpGqG of class D-modules on a reductive group G (D-module analogues of class functions). This rich category is the home of Harish-Chandra’s system of differential equations (recast is the language of D-modules by Hotta and Kashiwara), a setting for Springer Theory, and for Lusztig’s character sheaves (in their D-module interpretation due to Ginzburg). My ongoing Springer theory project [7, 10, 11] utilizes categorical and geometric techniques to bring surprising new results and a fresh understanding of these important works, shifting focus from the classical study of character sheaves to combinatorial descriptions of entire categories of D-modules. This work is set in the wider context of geometric Langlands in genus 1 and there are exciting applications to the geometric Langlands conjectures for elliptic curves [5], as well as Cherednik algebras and Hilbert schemes. Class D-modules also arise as the Drinfeld center of Harish-Chandra bimodules– a central player in the classical and categorical representation theory of reductive Lie groups. My work [1] on the quantum Ngoˆ action (which we show appears naturally in the context of Langlands dual Coulomb branches) explores this monoidal category from an entirely new perspective, with wide-ranging applications in categorical representation theory and mathematical physics. Topological field theory is a key tool in my work; we are able to extract hidden structure and symmetries from classical representation theoretic objects (for example in my work on the homology of character stacks [2], and on quantum D-modules).

1 In the remainder of this section, we will expand upon each of these themes in my research. 1.1 Parabolic Induction and Generalized Springer Theory. In Harish-Chandra’s phi- losophy of cusp forms, one studies objects (e.g. representations, class functions, sheaves) associated to a reductive group G by parabolic induction from analogous objects associated to certain smaller reductive groups (the Levi subgroups). Those objects which cannot be induced from any smaller Levi are called cuspidal. In my work [3] with David Ben-Zvi and Hendrik Orem we show (see Theorem 16) that there are no cuspidals for categorical representations of reductive groups: every (de Rham) G-category is contained in a principal series. On the other hand, in his remarkable 1984 paper [Lus84], Lusztig found (and classiﬁed) so-called cuspidal local systems, implying that the corresponding result fails for class D-modules (the home for characters of categorical representations). Motivated by Lusztig’s work, in my paper [11] I give a complete description of the abelian category of class D-modules on the Lie algebra (see Theorem 2), greatly extending Lusztig’s description for the nilpotent cone (the so-called generalized Springer correspondence).1 A number of mysteries remain surrounding the nature of the generalized Springer correspondence and the classiﬁcation of cuspidals. I am currently looking at generalized Springer theory from a new angle [6]: that of the underlying G-representation on global sections of the underlying D-module (inspired by the observation (see Section 2.4) that Hotta-Kashiwara’s results on the Harish-Chandra system characterize the “Springer block” as consisting of objects with an invariant vector). Beyond class D-modules on the group and Lie algebra already discussed, one may consider the following variants:

• Elliptic: replace G by the “elliptic group”, GE of framed semistable G-bundles on an elliptic curve E (see Section 2.7). • Quantum: replace G with the quantum group, diﬀerential operators with multiplicative diﬀerence operators (see Section 5.2).

• Mirabolic: In the case G “ GLn, adjoin a framing vector, and consider twisted diﬀer- ential operators (see Section 2.6). In each of these settings, one expects to have a generalized Springer theory package, with induction/restriction functors, character sheaves, cuspidal data, etc., and in each case there are a variety of applications. For example, mirabolic character sheaves give rise to modules for the spherical subalgebra of a Cherednik algebra via Hamiltonian reduction; a better understanding of generalized Springer theory in this setting would allow one to see parts of the category which are invisible to Hamiltonian reduction.

1Interestingly, as shown in my paper [10] while the block decomposition indexed by cuspidal data contin- ues to hold for the derived category, the Springer-theoretic description of the blocks in terms of relative Weyl groups fails (see Theorem 4). This forms an interesting counterpoint with the work of Rider who proves a derived Springer theorem for the nilpotent cone.

2 . 1.2 A Categorical Plancherel Theorem and the quantum Ngoˆ action. The classical Plancherel theorem of Harish-Chandra is a central result in harmonic analysis; it ex- presses a class function on G as an integral over the unitary spectrum G. In my paper [1] with David Ben-Zvi (as part of an ongoing joint project; see Section 3.4 below) we up- grade this result to the categorical level: expressing a class D-module onp G as a sheaf on the categorical spectrum (a certain form of the dual maximal torus modulo Weyl group symmetries T _{W ). Following this guiding principle has lead us through an extraordinary diversity of mathematics! The central player in our work the universal centralizer J, and its quantization, the category h of Whittaker D-modules. This is a fundamental object in representation W theory and gauge theory; in particular, it is the phase space for the Toda lattice and the Coulomb branch for 3-d pure “ 4 gauge theory. The universal centralizer also carries N a universal action on Hamiltonian G-spaces, as described in work of Knop and used in Ngo’sˆ proof of the fundamental lemma. As part of our work we show (see Theorem 17) that the convolution of bi-Whittaker D-modules is commutative, and construct a central action of h on G-categories: a quantum form of the universal action of J. Moreover, W the spectrum of the categorical commutative algebra h is the expected form of T _{W , W giving precise meaning to the categorical Plancherel theorem described above. Neither the commutativity nor the braided functor are manifest in the definition of Whittaker D-modules. To prove our results in [1], we utilize Langlands duality (namely, the derived geometric Satake theorem of Bezrukavnikov and Finkelberg). In this way we are able to express the various players described above in terms of the topology of the Langlands dual affine Grassmannian, where the desired structures are relatively appar- ent. The universal centralizer J and its quantum form h have appeared prominently in W recent work of Braverman-Finkelberg-Nakajima, Teleman, and Ginzburg, amongst other places, and the Ngoˆ homomorphism and its quantum form appears to be a key aspect of these stories which has not yet been fully explored. This work is still ongoing and there are many directions for future research. For example, we are thinking about: applications of the Ngoˆ action to the theory of G-categories, the role this structure plays in gauge theory, and potential ramifications in the more general setting of Knop’s work on spherical varieties. 1.3 Topological Field Theory and (Quantum) Character Varieties. From a mathematical perspective a topological field theory (TFT) is an assignment of algebraic invariants (e.g. categories, vector spaces, numbers) to smooth manifolds which satisfies certain rules with respect to cutting or gluing of manifolds. The motivating example is 2d gauge theory ZΓ for a finite group Γ ; this TFT assigns 1 Γ to a circle the space of class functions ZΓ pS q “ CrΓ s , and to a closed 2-manifold S, the

3 number ZΓ pSq of Γ -Galois covers of S (counted with multiplicity given by the reciprocal of the number of automorphisms). Unwinding the cut-and-paste laws for the TFT leads to a formula (originally due to Frobenius) for ZΓ pSq in terms of the characters of Γ . The case when the gauge group Γ is the symmetric group Sn is of particular interest; in this

setting the numbers ZSn pSq (and their variants for surfaces with punctures) are called Hurwitz numbers. In my paper [9] I construct an analogous TFT in order to extract a formula for so-called spin Hurwitz numbers (see Section 4.2), which compute Gromov- Witten invariants of certain complex surfaces amongst other things. In my recent work with Arun Debray [4], we construct an extension of the invertible “Arf theory” underlying this story to pin´-manifolds. This has applications in condensed matter physics. In my work [2] I construct a categorical analogue of ZΓ which would assigns the category of class D-modules DpGqG to a circle and the homology of the moduli stack of G-local systems to a surface S (Theorem 24), vastly improving upon the expectations of my coauthors in their earlier work. These homology groups are closely related to the cohomology of character varieties which are the subject of fascinating conjectures of Hausel and Rodriguez-Villegas. Our work brings forth the prospect of a Frobenius type formula for the homology of the character stack in terms of the categorical representation theory of G (this idea has captivated me since my early graduate work). Carrying out this program is one of the primary motivations for our work on the Plancherel formula described above. The techniques I have developed for D-modules are uniquely positioned to apply in the q-analogue of the character theory developed by Ben-Zvi–Brochier–Jordan: a TFT built is built in a natural way (factorization homology) from the braided monoidal category of representations of the quantum group. It is in this setting that the formalism of TFT shows its full power: a single gadget which encodes the quantum group, DAHA, skein algebras, and knot invariants. These ideas open up a new world of research: quantum categorical harmonic analysis! Together with David Jordan, we are currently putting this dream in to practice, starting with quantum analogues of Springer theory and cuspidal local systems (see Sec- tion 5.2). My work on class D-modules provide the link between geometric and algebraic approaches necessary for this program. Progress is onerous in this setting without the geometric theory of D-modules and perverse sheaves to fall back upon. However, my theory of twisted Hamiltonian reduction functors and the microlocal structure of parabolic induction/restriction are the most promising tool in our quantum generalized Springer theory. Note. The following sections have been prepared as part of an anticipated grant proposal, and contain a detailed account of prior, current, and future research directions.

4 Contents

2 Generalized Springer Theory 6 2.1 Background: Generalized Springer Theory and Character Sheaves . . . . .6 2.2 Generalized Springer Theory for equivariant D-modules on g ...... 6 2.3 Derived Springer theory ...... 8 2.4 Generalized Hamiltonian reduction ...... 9 2.5 The group setting and other variants: elliptic, quantum, mirabolic . . . . . 10 2.6 The mirabolic setting ...... 10 2.7 The elliptic setting ...... 12

3 Categorical representation theory and quantum Hamiltonian G-spaces 12 3.1 Categorical representations ...... 12 3.2 Regular centralizers and the Ngoˆ homomorphism ...... 13 3.3 Quantizing the Ngoˆ action via Derived Satake ...... 14 3.4 Spectral decomposition and quantum integrability ...... 15 3.5 The nil-DAHA and very central D-modules ...... 16

4 Topological ﬁeld theory and character theory 17 4.1 Background: ﬁnite group gauge theory and Hurwitz numbers ...... 17 4.2 Arf-(Brown) theory and Spin Hurwitz numbers ...... 17 4.3 Character Theory TFT and the cohomology of character stacks ...... 18

5 Quantum Character Theory 18 5.1 Background ...... 18 5.2 Quantum Generalized Springer Theory ...... 19 5.3 Connection with the elliptic setting ...... 20 5.4 Quantum G-categories and the multiplicative Ngoˆ action ...... 20 Conventions and notation. Throughout this research statement, G will denote a reductive algebraic group over the complex numbers, and we will ﬁx the following associated paraphernalia:

• B a Borel subgroup of G, N “ rB,Bs the unipotent radical of B, and H “ B{N the universal torus (when necessary we choose a splitting in to B, realizing T as a maximal torus in G) • g,b,n,h the corresponding Lie algebras.

• W the Weyl group (which we will sometimes identify with NGpHq{H for a maximal torus H)

If X is an algebraic variety or stack over the complex numbers, MpXq (respectively DpXq) will denote the abelian (respectively derived) category of D-modules on X.

5 2 Generalized Springer Theory 2.1 Background: Generalized Springer Theory and Character Sheaves. In his seminal paper [Lus84] Lusztig defined and classified cuspidal local systems and proved the generalized Springer correspondence, which classifies equivariant perverse sheaves on the unipotent cone of G. He then went on to develop the theory of character sheaves, and proved that an irreducible character sheaf is a summand of a parabolic induction from a cuspidal local systems. These ideas were recast in the D-module setting by Ginzburg [Gin89, Gin93] and Mirkovic [Mir04] leading to many simplifications. More recent work of Rider and Russell explore the derived direction of generalized Springer theory [Rid13, RR16]. One aspect of my approach which is fundamentally new, is that my results concern all equivariant D-modules, not just character sheaves (in particular, not just holonomic D-modules). The theory of character sheaves (as reformulated by Ginzburg) is about equivariant D-modules on G for which the Harish-Chandra center Zg acts locally finitely; thus character sheaves are discrete over the space SpecpZgq of central characters - character sheaves with incompatible central characters don’t “talk” to each other. Our understanding of the theory is greatly enhanced by being able to consider universal objects (like the universal Harish-Chandra system), clarifying the connection between parabolic induction/restriction, Hamiltonian reduction, and Cherednik algebras. Another major component of my work is considering this problem in a variety of new settings (elliptic, quantum, Mirabolic), each with its own interesting applications. My work also differs considerably in the methods and tools used. One example of this is the change in focus from the endomorphism algebra of induced objects (as studied in Springer theory) to the entire monad associated to the induction-restriction adjunc- tion. The problems proposed require both traditional abelian category methods, as well as more modern 8-category theoretic tools (such as the Barr–Beck–Lurie theorem). In many cases, I find various proofs of a given statement (for example, the exactness of parabolic restriction), each of which has the merit of being applicable in a different setting. For example, one requires a purely algebraic approach to deal with the quantum setting, whereas the elliptic setting (being non-affine) necessitates the use of geometry. 2.2 Generalized Springer Theory for equivariant D-modules on g. In my paper [11] I study the abelian category MpgqG of equivariant D-modules on g. This is the first example in a series of variations, described in later sections (see Sections 2.6 and 2.7). The key tools are the functors of parabolic induction and restriction

G G . L G ResP ,L : Mpgq n Mplq : IndP ,L, deﬁned for each Levi subgroup L of G. G G Deﬁnition 1. An object M P Mpgq is said to be cuspidal if ResP ,L M » 0 for every proper parabolic subgroup of G.A cuspidal datum is a pair pL, q consisting of a Levi subgroup L E of G and a simple cuspidal object of MplqL supported on the nilpotent cone in l “ LiepLq.

6 The main results of loc. cit. are summarized as follows: Theorem 2 ( [11], Theorems A, B, C). 1. The parabolic induction and restriction functors are exact and satisfy a natural Mackey formula.

G G 2. The category Mpgq decomposes as an orthogonal sum of subcategories MpgqpL, q indexed by conjugacy classes of cuspidal data. E

G WG,L 3. Each of the blocks MpgqpL,Cq is equivalent to Mpzplqq , or equivalently, modules for 2 the ring Dzplq#WG,L. Here zplq denotes the center of the Lie algebra l, which carries an action of the relative Weyl group WG,L :“ NGpLq{L (by results of Lusztig, WG,L is known to be a Coxeter group acting by reﬂections).

4. I deﬁne a partition of the variety of commuting elements in g:

commpgq “ commpgqpLq Lğ{„ and show that the objects which are sent to zero by parabolic restriction to a Levi L are characterized by the property that their singular support is contained in commpgqępLq. Thus the coarser decomposition of MpgqG in to the blocks for each Levi L is characterized entirely in terms of singular support.3 Restricting to the subcategory Mp qG of equivariant D-modules supported on the NG nilpotent cone we obtain an equivalence

G WG,L Mp Gq » Mpptq » ReppWG,Lq N pL, q pL, q àE àE Lusztig’s generalized Springer correspondence [Lus84] is the resulting bijection on simple objects. Here, simple objects of Mp qG are in bijection with pairs p , q consisting of a NG O L nilpotent orbit and a simple equivariant local system on (every object of Mp qG is L O NG necessarily regular holonomic, so Mp qG is equivalent to the corresponding category of NG equivariant perverse sheaves by the Riemann-Hilbert correspondence). Further restricting to the Springer block corresponding to the unique cuspidal datum for L “ T a maximal torus, we recover the (non-generalized) Springer correspondence. The proof of each individual part of Theorem 2 uses new ideas. For example, the proof of part 3 involves studying the monad aﬀorded by the parabolic induction and restriction

2This ring is an example of a rational Cherednik algebra with the parameter c “ 0. More general values of the c parameter appear in the mirabolic variant (at least in type A); see Section 2.6. 3This result suggests a close relationship with work of McGerty–Nevins [MN14] showing that certain stratiﬁcations of the cotangent bundle of a stack give rise to a recollement on the category of D-modules - in this case, the recollement turns out to be orthogonal.

7 functors; this is a novel approach to Springer theory which both elucidates the earlier approaches of Kazhdan-Lusztig, Ginzburg and others, and allows for substantial general- izations. The proof of exactness of parabolic restriction also introduces a new technique based on the idea that exactness holds “microlocally generically”, which opens the pos- sibility of generalizing to more general quantizations (such as quantum D-modules, see Section 5). 2.3 Derived Springer theory. In my paper [10] I show that there is an orthogonal decomposition on the derived category DpgqG, extending the generalized Springer decomposition in the abelian category. However, unlike the abelian category setting, there is no Mackey decomposition of the composite of induction and restriction–only a ﬁltration.

Theorem 3 ( [10]). The Mackey ﬁltration on the composite of parabolic induction and restriction is non-split already for G “ SL2. This has an important consequence for the Springer theoretic description of the blocks:

Theorem 4. For each cuspidal datum pL, q together with a choice of parabolic P containing L E as a Levi factor, there is a dg-ring A “ ApP ,L, q controlling the corresponding block: E G DcohpgqpL, q » A ´ Perf E Moreover, there is a ﬁltration on A (indexed by the Coxeter poset of the corresponding relative Weyl group) such that the associated graded is the following “triple aﬃne Hecke algebra”:

˚ Dzplq b Sympzplq r´2sq #WpG,Lq ` ˘ This ﬁltration is non-split in general (e.g. for the Springer block in the case G “ SL2). What is notable about this result is that the corresponding ﬁltrations on the algebra controlling the following two subcategories do split, leading to the following characteri- zations:

• The abelian category blocks are controlled by Dzplq#WG,L (by my work [11]). • The blocks of the equivariant derived category on the nilpotent cone are controlled by Sympzplqqr´2sq#WG,L (by work of [Rid13, RR16]). Theorem 4 implies that one cannot combine these results to get a description of the entire block of DpgqG. This naturally leads to the following problem (already interesting in the case G “ SL2): Problem 5. Give a combinatorial description of the (generalized) Springer blocks of DpgqG.

8 2.4 Generalized Hamiltonian reduction. In [6], I show how my theory of parabolic induction/restriction together with results of Hotta-Kashiwara [HK84] explain the close relationship between Quantum Hamiltonian reduction and parabolic restriction/induction to the torus.

Theorem 6. The Hamiltonian reduction D-module Mg,C “ Dg{G is a projective generator of G W the Springer block; under the generalized Springer correspondence MpgqSpr » Mptq , it corresponds to the object Dt (with its canonical W -equivariant structure). In particular the Springer block is characterized by the property that any non-zero object has a G-invariant vector. As a corollary, we obtain a new proof of the Harish-Chandra isomorphism of Levasseur– Staﬀord [LS95, LS96]:

G W Dg{G » EndMpgq Dg{G » EndDh#W pDhq » Dh

I am currently working` on˘ a new proof` of Theorem˘ 6 which doesn’t depend on Hotta- Kashiwara (and thus can be used to give a new proof of their results). This suggests the following:

Problem 7. What is the analogue of the Hotta-Kashiwara characterization of the Springer block for other blocks (corresponding to cuspidal data)? In other words, how can the generalized Springer theorem be understood in terms of the global sections of the corresponding D-modules as G-representations.

In [6], I suggest and prove some partial solutions. We deﬁne generalized quantum Hamiltonian reduction D-modules, Mg,V :“ Dg bUg V associated to a representation V of G. These form a set of compact, projective generators of MpgqG.

Problem 8. Compute the image of Mg,V under the generalized Springer correspondence. I also show that the global sections of a simple cuspidal D-module on a semisimple group G is admissible as a G-representation (every irreducible representation appears with ﬁnite multiplicity), which leads to the following:

Problem 9. Compute the character of the admissible G-representation associated to a cuspidal local system. I expect these ideas will shed some light on Lusztig’s still mysterious classiﬁcation of cuspidal local systems in [Lus84], as well as the cleanness property. For example, (as shown in my paper [6]) in the case G “ SLn the blocks of the generalized Springer decomposition are entirely determined by central character which can be read oﬀ from the underlying G-representation on global sections. In particular, the objects M k n are cuspidal whenever k is relatively prime to n, giving a new presentation sln, C of cuspidal local systems in type . Ź A

9 2.5 The group setting and other variants: elliptic, quantum, mirabolic. There are a number of variations on the category of equivariant D-modules on g for which the generalized Springer theory I have developed might be extended. For example, the most natural example is the category MpGqG of conjugation equivariant D-modules on G; we will refer to this as the group setting. In fact, understanding the this category of “class D-modules” on the group was the original motivation for this project. In many ways, the group setting looks similar to the Lie algebra setting. However, the Fourier transform for D-modules is a key technique in [11] which is not available in the group setting. In forthcoming work [7] I develop new techniques which will allow us to prove the analogue of Theorem 2 in the group setting.4 The Lie algebra and group settings (or equivalently the stacks g{G and G{ad G) are the first two parts of a trichotomy, the third of which is the elliptic setting. Here, one 0,ss studies the category of D-modules on the moduli stack BunG pEq of degree 0 semistable G-bundles on an elliptic curve E. Letting the smooth elliptic curve degenerate to a genus 1 curve with a single node (respectively, cusp) recovers the group setting (respective, Lie algebra setting).5 I will discuss the elliptic setting further in Section 2.7. q Another variation arises from replacing differential operators DG by DG, an appro- priate version of multiplicative difference operators on G; this is referred to as the quan- ˚ q tum setting. Just as DG is a quantization of T G » G ˆ g, DG is a quantization of the multiplicative cotangent bundle G ˆ G (with its Poisson structure as given by Semenov- Tyan-Shanskii [STS94]). This quantization is closely connected with the quantum group. Further background and research directions are given in Section 5. Finally, for G “ GLn in each of these settings, there is natural deformation of these categories which relates to Cherednik and double affine Hecke algebras of various kinds. Research in this area is outlined in Section 2.6 below.

2.6 The mirabolic setting. In the case G “ GLn, the category of equivariant D-modules on g admits a natural family of deformations, namely, twisted equivariant D-modules on n X :“ gln ˆ C . One motivation for considering these mirabolic D-modules is that the functor of quantum Hamiltonian reduction takes values in the category of modules for the spherical subalgebra Uc of the rational Cherednik algebra (deforming the algebra of W invariant Dh appearing in Section 2.4). These ideas have been the subject of considerable interest, for notably in the work of Ginzburg with Bellamy, Etingof, Finkelberg, and Gan [BG15, EG02, FG10b, GG06]. It is natural to expect a pattern as in [11]: there is a functor from Mirabolic D-modules to the full Cherednik algebra Hc, deforming the functor to Dh#W deﬁned by parabolic restriction. Parabolic induction and restriction functors have been deﬁned in related settings (at

4I have recently learned of a paper of Bezrukavnikov and Yom-Dim in which the exactness of parabolic induction and restriction is proved in the group setting, via a diﬀerent approach. 5 The group of degree 0 line bundles on the cusp, respectively node, respectively elliptic curve E, is Ga, respectively, Gm, respectively E; this is a familiar trichotomy which is pervasive in mathematics.

10 least for c “ 0) by Finkelberg–Ginzburg–Travkin [FGT09] and Shoji–Sorlin [SS14]. How- ever, in this case, for each parabolic in gln given by a partial flag of k steps, there are k different functors. Thus the shape of Theorem 2 looks substantially different in this case, and must be appropriately modified. It is also not entirely clear (to me, at least) what the precise relationship is between these functors and parabolic induction and restriction functors of Bezrukavnikov–Etingof [BE09] or the specialization of Bellamy–Ginzburg [BG15]. Another way in which this situation may look different is that the decomposition by cuspidal data may not be orthogonal in general.

G Problem 10. Study the generalized Springer partition of McpXq aﬀorded by by induction and restriction functors, and establish a recollement situation.

Conjecture 11. The generalized Springer recollement is characterized by singular support, and is compatible with the recollement of McGerty–Nevins given by the Kirwan–Ness stratiﬁcation.

The characterization of the decomposition of DpgqG by singular support gives evi- dence for this conjecture.

Conjecture 12. The “associated graded” blocks of the recollement are given by modules for rational Cherednik algebras of relative Weyl groups (in this case all of type A).

Problem 13. Describe the category of mirabolic character sheaves (for example, study it’s highest weight theory in relation to category of the Cherednik algebra). O Previous work in this direction has focused on the Hamiltonian reduction formalism; I expect that the parabolic induction and restriction package will lead to a natural solution to these problems. 2 The Hilbert Scheme, HilbnpC q, is a symplectic quotient of G acting on an open sub- ˚ G set of T X. Thus a localization of the category McpXq is a quantization of the Hilbert scheme, as explained by Kashiwara–Rouquier [KR08]. There is a famous vector bundle 2 on HilbnpC q called the Procesi bundle, which formed a key part of Haiman’s celebrated proof of the n!-conjecture and the positivity of Macdonald polynomials. Understanding this collection of ideas using techniques of quantization and D-modules has been the source of much recent work, including Losev [Los14] and Ginzburg [Gin12] (see also the survey of Losev [Los15]).

2 Problem 14. Give a geometric construction of the Procesi bundle on HilbnpC q in terms of a universal Springer sheaf, and use it to deduce geometric properties of the isospectral Hilbert scheme.

The techniques I have developed in [11] pave the way to a better understanding these important ideas.

11 2.7 The elliptic setting. Given an elliptic curve E and a reductive group G, let GE denote the moduli stack of semistable, degree 0, G-bundles on E. The stack GE can be written as a quotient GE{G, where GE is a variety which parameterizes bundles as above, but with a framing at the identity element of E. In forthcoming work with Dragos Fratila and Penghui Li [5], we make the connection between the geometry of GE and that of G and g precise, by exhibiting a Lusztig stratification for GE in which the strata are expressed in terms of “psudo-psuedo-Levi subgroups” M (the joint stabilizer of a pair of semisimple elements in G). The starting point for these ideas is Atiyah’s classification of vector bundles on E [Ati57], with subsequent work by Friedman–Morgan–Witten [FMW98], and Baranovsky– Ginzburg (later with Evens) [BG96] [BEG03]. More recent work in this direction has been done by Ben-Zvi–Nadler [BZN15], Li–Nadler [LN15], and Fratila [Fra16]. It was shown by Looijenga (unpublished, see [BG96]) that there is an analytic isomorphism between BunGpEq and a twisted form of the adjoint quotient for the loop group LG, where the twist depends on the modulus of E; Li and Nadler [LN15] prove that the category chpGEq of elliptic character sheaves is independent of the modulus. This justifies the assertion that chpGEq (or possibly more generally on all of BunGpEq) is a model for character sheaves on LG, and thus a part of local geometric Langlands. On the other hand, considering GE as the semistable locus in BunGpEq, puts the theory of D-modules on GE in the setting of global geometric Langlands.

Conjecture 15. The natural analogue of Theorem 2 holds for the category of elliptic class D- modules pGq “ MpG qG. In particular, elliptic character sheaves admit a generalizer Springer C E description in terms of representations of elliptic Weyl groups.

Considering the mirabolic variant in the elliptic setting leads to a connection with elliptic Cherednik algebras (a special case of a construction of Etingof [Eti04], see also Finkelberg–Ginzburg [FG10a]). One may also hope to extend these ideas to deeper Harder– Narasimhan strata, and to nonzero degree; exciting work of Fratila [Fra16] suggests a beautiful approach in this direction. 3 Categorical representation theory and quantum Hamiltonian G-spaces The projects in this section are centered around the notion of a smooth (also called de Rham, or strong) G-category: a module category for the monoidal category DpGq of D- modules on G under convolution. Examples of smooth G-categories are g ´ mod and DpXq for any G-space X. 3.1 Categorical representations. Let HC denote the monoidal category of Harish-Chandra bimodules, and the universal Hecke category DpNzG{Nq.6 In my work [3] with Ben-Zvi H 6A monodromic variant of this category involves additionally taking weak H-invariants on each side.

12 and Orem, we prove that G-categories satisfy an analogue of Casselman’s theorem: every G-category appears in a principal series. Theorem 16 ( [3]). The functor of strong N invariants is conservative, and deﬁnes a categorical Morita equivalence between the monoidal categories DpGq and . H Combining this with a result of Beraldo [Ber13] gives:

Smooth G-categories o „ / -module categories o „ / HC-module categories

P H P P DpG{Kq o / DpNzG{Kq o / pg,Kq ´ mod These equivalences enhance the usual theory of Beilinson–Bernstein localization (see also the work of Ben-Zvi and Nadler [BZN12]), and they provide a bridge between the geometric theory of G-categories and the well studied theory of Harish-Chandra bimodules, finite Hecke categories, Koszul duality, and Soergel-bimodules, (see e.g. Soergel [Soe92], Beilinson–Ginzburg–Soergel [BGS96], Bezrukavnikov–Yun [BY13], Elias–Williamson [EW14]). 3.2 Regular centralizers and the Ngoˆ homomorphism. My ongoing project with David Ben-Zvi (originating in discussions with David Nadler during my doctoral studies) is based on the following situation. The universal centralizer is a smooth group scheme (i.e. fiberwise algebraic group) over ˚ c, whose fiber Ja is the centralizer in G of a regular element x P g with χpxq “ a (one can choose a regular element x for each a by means of the Kostant slice κ : c Ñ g˚). The group scheme J (its total space, that is) is moreover a symplectic affine variety. As such it admits ˚ another description via Hamiltonian reduction: J » NψzzT G{{ψN, where ψ is a generic character of N. Another key perspective on J comes from the work of Bezrukavnikov– Finkelberg–Mirkovic [BF08]; they show that the equivariant homology of the Langlands dual affine Grassmannian recovers the coordinate ring of J.7 As explained by Knop [Kno96] and Ngoˆ [Ngo10],ˆ J’s role in life is to integrate Hamil- tonian flows: for any Hamiltonian G-space M with moment map µ : M Ñ g˚ the commuting Hamiltonians induced by the Poisson-invariant map M Ñ g˚ Ñ c integrate to an a fiberwise action of J. These actions are implemented by a universal action, which is best understood as a Lagrangian correspondence

˚ η ˚ J o g ˆc J / T G (1) where the map η is a homomorphism into the group scheme of centralizers I Ď T ˚G (consisting of elements of the cotangent bundle whose left and right invariant projections to g˚ agree). 8

7We now understand this as a special case of a Coulomb branch (for the pure gauge theory) by the work of Braverman-Finkelberg-Nakajima [BFN16]. 8As observed by Ngoˆ [Ngo10],ˆ the existence of η is a simple consequence of the Hartog principle: it is the extension of a tautological map deﬁned on the regular locus of g˚, whose complement has codimension at least 3.

13 3.3 Quantizing the Ngoˆ action via Derived Satake. The starting point for this project is our construction in [1] of a quantization of J as an abelian group scheme and of the Ngoˆ homomorphism. Our construction deﬁnes a universal action of the quantization of J on any quantum Hamiltonian G-space (i.e. de Rham G-category), giving meaning to the notion of Langlands parameters for categorical G-representations.

Theorem 17 ( [1]). There exist:

• A co-commutative Hopf algebroid Wh~ (or alternatively its symmetric monoidal category of modules h ), quantizing J. W ~ • A braided monoidal functor

Ngoˆ : h Ñ D pGqG ~ W ~ ~ quantizing the Lagrangian correspondence (1).

The proof of Theorem 17 stems from our interpretation of the Ngoˆ homomorphism under Langlands duality. The starting point is the derived geometric Satake of Bezrukavnikov- Finkelberg [BF08], which deﬁnes an equivalence

˘ DpGrq » HC~

Here, the left hand side is the (renormalized) category of D-modules on the aﬃne Grass- _ _ _ mannian quotient stack Gr “ L`G zLG {L`G ; the right hand side is the category of (diﬀerential graded) Harish-Chandra bimodules, i.e. G-equivariant modules for the Rees algebra U~g (the parameter ~ corresponds to loop rotation on the left hand side, thus appears in cohomological degree 2). Moreover, the authors explain that the natural module _ category D˘ ppt{L`G q for D˘ pGrq corresponds to the quantum Kostant slice (i.e. Kostant- Whittaker reduction) on the right hand side. _ _ _ It is helpful to abstract this situation: the double quotient L`G zLG {L`G is a groupoid acting on BL G. More generally given any groupoid Ñ X, and suitable sheaf ` G theory Shvp´q we have a corresponding Hecke category “ Shvp q. H G Theorem 18 ( [1]). The category ShvpXqG carries a canonical symmetric monoidal structure and there is a braided monoidal functor to the Drinfeld center

ShvpXqG Ñ p q Z H splitting the natural action functor in the opposite direction.

Applying this result in the case of the aﬃne Grassmannian leads to Theorem 17.

14 3.4 Spectral decomposition and quantum integrability. One of the fundamental problems in harmonic analysis is spectral decomposition of functions on a symmetric space under Harish-Chandra’s commutative algebra of invariant differential operators, a collection of higher analogs of the Laplace operator for which we seek joint eigenfunctions. We now describe some consequences of Theorem 17 in this setting; understanding and exploiting these consequences is one of the primary directions of current and future research. By a result of [Gai15, Ber13], module categories for HC are identified with DpGq- modules, also known as de Rham or strong G-categories. The theory of de Rham G- categories, or the equivalent theory of HC-modules, is a natural realization of the notion of quantum Hamiltonian G-space (an algebraic variant of an idea of [Tel14]). For a G-space M, there is a natural family of commuting G-invariant differential operators (the Casimir operators), which provides a source of many quantum integrable systems [Eti07]. In particular given λ P c » h˚{{W we can define the λ-eigensystem for the Harish-Chandra Laplacians in this setting; the quantum analog of the fibers of the classical Hamiltonians χ ˝ µ. However, unlike in the classical setting, quantum Hamilto- nian G-spaces do not “live” over c “ SpecpZgq: there is no spectral decomposition of M over c. On the other hand, Theorem 17 gives a spectral decomposition of quantum M Hamiltonian G-spaces over “Spec h2, which, by a result of Lonergan [Lon18] (in the W case G is of adjoint type) one can identify with H_{{W .9

Corollary 19 ( [1]). For any D pGq-module , there is an action of the tensor category h œ ~ M W ~ commuting with the D pGq action. M ~ For example, class D-modules DpGqG form a sheaf of braided monoidal categories ˚ aff 10 h,rλs over h {{W . Thus we have a braided monoidal category DpG{ad GqW , a refined (or strict) version of the category DChrλspGq generated by character sheaves with central character rλs. Similarly DpGq, , and HC all admit a spectral decomposition as monoidal H categories over h˚{{W aff.

Problem 20. Compute the fibers of the monoidal categories DpGq, , DpGqG, and HC with H respect to the spectral decomposition afforded by the Ngoˆ action. Remark 21. It appears that there is a close connection with work of Bezrukavnikov– Finkelberg–Ostrik [BFO09,BFO12] on certain multi-fusion categories, which are constructed from and HC (at a fixed central character) by taking the associated graded for a certain H filtration indexed by two sided cells. 9More precisely, Wh is identified with the spherical nil-DAHA (see Section 3.5 below), and its modules may be thought of as sheaves on h˚ which descend to the coarse quotient h˚{{W aff. This analytically (though not algebraically) identified with H_{{W . 10In particular, any class D-module localizes over h˚{{W aff; this is what we referred to as the categorical Plancherel Theorem in the overview, Section 1.2

15 3.5 The nil-DAHA and very central D-modules. Comparing our work with the recent literature on Whittaker differential operators and the nil-DAHA (in particular, work of Ginzburg [Gin] and Lonergan [Lon18]) yields some pleasing consequences, which we plan to explore in forthcoming work. First let us consider a concrete interpretation of the Ngoˆ functor, as a DG ´ Wh- bimodule structure on B :“ Ug bZg Wh (quantizing the Lagrangian correspondence 1). By the theory of Hamiltonian reduction and the Harish-Chandra homomorphism (see 2.4), the G-invariants G G B “ Ug bZg Wh » Wh W have a left action of pDH q commuting with the right regular Wh-action; in other words, W the Ngoˆ functor determines a ring homomorphism DH Ñ Wh. Moreover, at the level of abelian categories, the functor of G-invariants is “almost” conservative; thus in this setting the ring homomorphism above determines the Ngoˆ homomorphism as a plain functor (crucially, however, one cannot see the monoidal structure in this way). W A homomorphism DH Ñ Wh can already be found in the literature—at least in the case when G is of adjoint type. Ginzburg [Gin] and Lonergan [Lon18] (by very different sph methods) exhibit an isomorphism between Wh and the spherical subalgebra of the AW aff so-called nil-DAHA AW aff (the affine nil-Hecke algebra associated to the affine Weyl group sph of G); the ring DW naturally sits as a subring of giving rise to the expected homo- H AW aff ˚ morphism. Modules for the nil-DAHA AW aff should be thought of as sheaves on h with descent data to the coarse quotient h˚{W aff (this quotient by an infinite discrete group is not a conventional object of algebraic geometry; note that the C-points are in bijection with H_{{W , but the algebraic structure is very different). As we expect to show in a sequel to [1], the two homomorphisms (one coming from the Ngoˆ functor, the other from the nil-DAHA and Hamiltonian reduction) agree. This has the following consequence:

Conjecture 22. The composite functor

W G AW aﬀ ´ mod ãÑ DpHq Ñ DpGq carries a natural braided monoidal structure.

What makes this statement remarkable is that the functor DpHqW Ñ DpGqG aﬀorded by quantum Hamiltonian reduction/Springer theory11 does not carry any monoidal structure at all unless one restricts the the subcategory of nil-DAHA modules. The monoidal properties in Conjecture 22 follow from another conjecture concerning very central D-modules: objects of DpGqG whose image under the horocycle transform G NzG{N NzB{N DpGq Ñ D H is supported on the diagonal H . This vanishing condition was ´ ¯ 11See Section 2.4 for further explanation.

16 defined in my PhD thesis [8], and has been considered in the recent work of Tsao-Hsien Chen [Che16]. In my thesis, I predicted the following relationship between very central D-modules and sheaves on the coarse quotient: Conjecture 23. The essential image of the composite functor W G AW aff ´ mod ãÑ DpHq Ñ DpGq consists of very central D-modules. For each W aff orbit χ in h˚, the Ngoˆ functor defines an object in DpGq. Conjecture Eχ 23 predicts these objects to be interesting twisted forms of the Harish-Chandra systems, whose behavior under convolution looks as if they were skyscraper sheaves under tensor product. I recently learned that these objects have been considered by Chen [Che16] under the name , where the author notes some “remarkable” properties. These prop- Mθ erties are explained, and put in a wider context by our work. 4 Topological field theory and character theory 4.1 Background: finite group gauge theory and Hurwitz numbers. Gauge theory for a finite group Γ provides a useful toy model for these ideas. This theory counts the number ZΓ pΣq of Γ -Galois covers on a closed surface Σ. Analyzing how the surface can be built up by cutting and gluing pairs of pants gives rise to a formula for these numbers (first obtained by Frobenius [Fro68]) in terms of the representation theory of Γ : χpΣq ZΓ pΣq “ pdimV {#Γ q , (2) VÿPΓ where Γ is the set of irreducible representationsp of Γ .

4.2 Arf-(Brown) theory and Spin Hurwitz numbers. In the case Γ “ Sn, the invariants of the finitep group gauge theory are known as Hurwitz numbers. Spin Hurwitz numbers are a variant which counts covers of a spin surface with a sign according to the parity (also called Atiyah invariant, or Arf invariant) of the covering surface. These numbers are useful for the Gromov–Witten theory of complex surfaces (see [MP08, LP09, LP13] as well as [EOP08] for a different appearance). In my paper [9] I construct a TFT which computes Spin Hurwitz numbers, and use it to give a combinatorial expression for them (see also [Lee18] for a proof using analytic techniques, and [EOP08] for a proof in the genus 0 case). The base case of this theory is the so-called Arf theory, a certain invertible TFT which assigns ˘1 to a closed spin surface according to its parity. In my paper [4] with Arun Debray, we study an extension of the Arf theory (the Arf-Brown theory) to pin´ manifolds. This work was inspired by recent progress in condensed matter physics on the classification of topological phases in terms of their associated low energy theory, in particular the work of Freed and Hopkins [FH] (see also [GJF] for further references); the Arf-Brown theory is connected with the a certain physical system called the Majorana chain.

17 4.3 Character Theory TFT and the cohomology of character stacks. Given a topological space X (assumed connected for simplicity), the character stack

pXq :“ Hompπ pXq,Gq{G, MG 1 parameterizes G-local systems on Σ. In joint work with Ben-Zvi and Nadler, we show:

Theorem 24 ( [2]). There is a topological field theory (TFT) whose value on a closed XG surface Σ is the Borel-Moore homology H p pΣqq. ˚ MG One can think of the TFT G as a categorical version of 2d finite group gauge theory X 12 ZΓ described in Section 4.1 above. Hausel and Rodriguez-Villegas compute certain Hodge theoretic invariants of a closely related character variety in terms of the GpFqq-Hurwitz numbers, which in turn can be understood in terms of Lusztig’s character sheaves. We seek to obtain a direct relationship between the topology of pΣq and the geometric character theory of D-modules, or in MG other words, to find a categorical version of the Frobenius formula. The following conjecture is a starting point for this goal; it concerns how to express the homology of the character stack as an “integral” over the space of categorical characters Spec h (see Section 3.4). W Conjecture 25. The TFT Z can be enhanced to a h-linear TFT, or in other words, a family G W of TFT’s over Spec h. In particular, H p pΣqq localizes as a sheaf over h˚{{W aff. W ˚ MG This conjecture gives a precise way to relate the combinatorics of finite Hecke categories (which control the fibers over points in h˚{{W aff) with the topology of character varieties. In particular, when combined with Remark 21, would give a relation between the TFT (which computes cohomology of character varieties) and 3-d finite group gauge XG theory for Lusztig’s finite groups associated to 2-sided cells. It also suggest the following:

Problem 26. Use the Koszul duality of ﬁnite Hecke categories to deduce a Langlands duality statement for the Borel–Moore homology of the character stack.

Relating this to the conjectures of [HRV08] requires the development a mixed (Hodge- theoretic) analogue of the universal character theory . I believe that this will be acces- XG sible with the tools we are developing, and will eventually lead to a pleasing conceptual interpretation of the combinatorics described in [HRV08]. 5 Quantum Character Theory q 5.1 Background. The ring of quantum diﬀerential operators, DG is a certain deformation of the ring of functions on G ˆ G (in the same way that the usual ring of diﬀerential operators on G is a deformation of the ring of functions on T ˚G – G ˆ g). It has been

12Similar ideas have been studied the setting of unipotent groups in positive characteristic by Boyarchenko–Drinfeld [BD13].

18 studied extensively by Semenov-Tyan-Shanskii, Alekseev, Backelin–Kremnizer, and Jor- ˆ n q dan [Ale93,BK06,STS94,Jor09]. For example, if H » pC q is an algebraic torus, then DH is the ring of multiplicative difference operators on H. In the work of Ben-Zvi, Brochier, Jordan, and Snyder [BZBJ15, BZBJ16, BJS18] the q authors construct and study a TFT ZG whose value on a manifold M (of dimension at most 3) gives a quantization of the character stack of M. They show that the value on a 1 1 q G 2-torus S ˆ S is the category of strongly equivariant equivariant DG-modules MqpGq - a quantum analogue of the category of class D-modules. In particular, a knot complement G in a closed 3-manifold defines a certain object of the category MqpGq which the authors conjecture is related to the quantum A-polynomial and Jones polynomial of the knot. In ongoing work of myself with David Jordan, we are applying the tools developed in my generalized Springer theory [7, 10, 11] to study the category MqpGqG, and in particular to gain a better understanding of these knot and 3-manifold invariants. Throughout this section we assume generic values of the parameter q P Cˆ (i.e. avoiding roots of unity). 5.2 Quantum Generalized Springer Theory. In order to prove an analogous result to Theorem 2 in the quantum setting, one must first define functors of parabolic induction and restriction (this is not so straightforward, as we do not have the rich functoriality of D-modules available to us). Problem 27. Define functors

G G . L G ResP ,L : MqpGq n MqpLq : IndP ,L for each parabolic subgroup P of G with Levi factor L, and show that these functors are exact and satisfy a Mackey formula. One potential approach to this problem is to study the natural domain wall between q q the TFT’s ZG and ZL associated to the parabolic P ; by the cobordism hypothesis, such a structure is determined by its value on a point. Given a collection of functors as in Problem 27 one can then formulate a conjectural quantum generalized Springer theorem:

G Conjecture 28. The category MqpGq decomposes as an orthogonal sum in to blocks indexed by “quantum cuspidal data” pL, qq. Each block is equivalent to a category of quantum D-modules E on an algebraic torus equivariant for a certain ﬁnite group (the stabilizer of the cuspidal datum under the action of the relative Weyl group).

G In particular, the conjecture predicts that there is a Springer block of MqpGq which is W q equivalent to MqpHq “ DH ¸W ´mod, where H is a maximal torus with Weyl group W . G q Thus, given an object M of MqpGq (for example the object assigned by the TFT ZG to a knot complement), parabolic restriction to the torus produces a new object N of MpHqW which is far more concrete—it corresponds to a q-diﬀerence equation. Conjecturally (in

19 the case G “ SL2), the quantum A-polynomial of a knot complement defines a solution to the corresponding q-difference equation on the torus H. In an analogous fashion to the situation described in Section 2.4, the parabolic restriction to the torus functor will be closely related to the quantum Hamiltonian reduction functor (defined in [BZBJ16]), which takes values in modules for the spherical subalge- q W bra pDH q . More precisely, these functors will be equivalent via the Morita equivalence q between DH ¸ W and its spherical subalgebra. As shown in [BZBJ16], a mirabolic variant of this story leads to modules for the spherical subalgebra of the DAHA. An expected analogue of the theory outlined in Section 2.6 produces a direct relationship with the full DAHA, clarifying this circle of ideas. Another natural problem arising from Conjecture 28 is to classify the quantum cuspidal data for a given group G.13 Currently, the most promising tool for constructing cuspidal objects is the generalized Hamiltonian reduction technique developed in my paper [6] (modulo Problem 27, which must be solved to define the notion of cuspidal). 5.3 Connection with the elliptic setting. Recall that the elliptic setting concerns D- modules on the “elliptic group” GE (see 2.7). Although the quantum and elliptic settings are very different in definition, there are good reasons one might expect a close relationship. For example, a q-local system on a torus T is specified by the same data as a T -local ˆ Z system on the elliptic curve Eq “ C {q . Also (as explained in my forthcoming work with Fratila and Li [5]) the Lusztig stratification of the commuting stack of G is parameter- ized by the same combinatorics as that of the Lusztig partition of GE (or equivalently its cotangent bundle). These ideas suggest the following:

Conjecture 29. There is an equivalence of categories between quantum character sheaves (ob- G jects in MqpGq satisfying a natural local ﬁniteness condition, and elliptic character sheaves G (analogous objects in MpGEq q ). 5.4 Quantum G-categories and the multiplicative Ngoˆ action. The theory of categorical representations as described in Section 3 has a natural q-analogue: a weak (respectively strong) quantum G-category is a module category for QCqpGq :“ qpGq ´ mod (re- q O spectively DqpGq :“ DG ´ mod). Strong quantum G-categories quantize the notion of quasi-Hamiltonian G-spaces, just as strong G-categories quantize the notion of Hamilto- nian G-spaces. There is also a multiplicative14 analogue of the universal centralizer J and its quantization, the Whittaker D-modules h (see Section 3.2). One must replace the the Kostant W slice and (its quantum form) the -algebra with the Steinberg slice and the q´ algebra W W (see e.g. [Sev11]). According to [BFM05] the multiplicative universal centralizer and its

13 SL Based on the analogous situation for D-modules studied in [CEE09], cuspidal objects in MqpSLnq n should give rise to ﬁnite dimensional modules for the DAHA via the functor constructed in [Jor09]. 14Here, multiplicative refers to the fact that the Lie algebra g (or its dual) are replaced by the group G. Typically, the quantization of multiplicative objects involve the quantum group.

20 quantization can also be realized in terms of the equivariant K-theory of the Langlands dual aﬃne Grassmannian.

Conjecture 30. There is a symmetric monoidal category h of q-Whittaker D-modules which W q acts centrally on every quantum G-category.

Following the approach of our paper [1], suggests using a multiplicative analogue of the derived geometric Satake theorem of [BF08] (such an analogue has recently been considered by Cautis–Kamnitzer [CK18]). The theory proposed here in the multiplicative setting features rich structures which were not visible in the story of Section 3; in the language of supersymmetric gauge theory, there is an extra loop. In particular, the multiplicative form of J now carries an interesting cluster structure, see [Wil16]. I believe this exciting direction in my research will lead to a new understanding of quantum integrable systems and supersymmetric gauge theory amongst other things. Individual and joint work [1] David Ben-Zvi and Sam Gunningham, Symmetries of categorical representations and the quantum Ngoˆ action, arXiv preprint: 1712.01963.

[2] David Ben-Zvi, Sam Gunningham, and David Nadler, The character ﬁeld theory and homology of character varieties, To appear in Mathematical Resrarch Letters, arXiv preprint: 1705.04266 (2017).

[3] David Ben-Zvi, Sam Gunningham, and Hendrik Orem, Highest Weights for Categori- cal Representations, To appear in International Mathematical Research Notices, arXiv preprint: 1608.08273 (2016).

[4] Arun Debray and Sam Gunningham, The Arf-Brown TQFT of pin´ surfaces, To appear in Contemp. Math., arXiv preprint: 1803.11183 (2018).

[5] Dragos Fratila, Sam Gunningham, and Penghui Li, Nilpotent cones and a stratiﬁcation of the moduli stack of semistable G-bundles on an elliptic curve, In preparation.

[6] Sam Gunningham, Generalized quantum Hamiltonian reduction and cuspidal D- modules, In preparation.

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[8] Sam Gunningham, Categoriﬁed Harmonic Analysis on Complex Reductive Groups, PhD Thesis, Northwestern University (2013).

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[11] Sam Gunningham, Generalized Springer theory for D-modules on a reductive Lie algebra, Sel. Math. New Ser. (2018).

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