Research Statement. Anna Romanov

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1 Research Statement Anna Romanov My research is in the representation theory of Lie groups. Lie groups emerged in the late 19th century as a mathematical description of the symmetries of the physical world, and throughout the last century, their presence has been a unifying thread connecting algebra, analysis, geometry, and physics. Understanding Lie groups and their associated Lie algebras by examining their actions on vector spaces has been a particularly fruitful method for revealing their structure, and this is the approach that I use in my work. I study representations of Lie groups through two distinct methods: one method uses algebraic geometry to reveal combinatorial patterns underlying categories of representations, and a second method exploits certain subgroups to examine the representation theory using tools from analysis. My first and primary method lies within the scope of geometric representation theory. There is a powerful connection between representations of Lie algebras and differential equations on algebraic varieties due to Beilinson and Bernstein [BB81]. This bridge lets us translate questions about representation theory of Lie algebras into questions about D-modules on flag varieties, which we can then approach using the robust machinery of algebraic geometry. My dissertation uses this technique to show that the multiplicities of certain irreducible representations (irreducible Whittaker modules) in a class of induced representations (standard Whittaker modules) are determined by a collection of polynomials which are defined geometrically. This result establishes a combinatorial skeleton of the category of Whittaker modules and leads to other natural structural questions about this class of representations. This is the main research program I intend to pursue in the future. The second method I use to study Lie groups draws on tools from analysis. From an irreducible representation of a Lie group G, one can construct a spherical function by taking the matrix coefficient of a vector in the representation that is fixed by a certain Lie subgroup K. In the setting of the orthogonal group, this process leads to the field of spherical harmonics. In general, this process leads to the notion of a Gelfand pair (G, K). Understanding the topological structure of the space of spherical functions associated to a Gelfand pair gives insight into the irreducible representations of the group G. My second project, which is collaborative work that stemmed from an AMS Mathematical Research Community workshop in June 2016, develops a new topological model for the space of spherical functions associated to a certain class of nilpotent Gelfand pairs. I plan to develop this project into a secondary research program which could include undergraduate researchers. Each of these projects opens up new unanswered questions in representation theory. In the following pages, I outline the progress I have made in each project and lay out specific future directions in which I plan to take the work. 1 A Kazhdan-Lusztig algorithm for Whittaker modules 1.1 Motivation A fundamental goal in representation theory is to classify of all irreducible representations of a Lie algebra. Such a classification is a daunting task, and it has only been done completely for the 1

2 simplest example: the Lie algebra sl(2, C) [Blo81]. One way to make this task more manageable is to study subcategories of representations subject to certain restrictions, then work to relax the restrictions and expand the categories. Block s classification of the irreducible representations of sl(2, C) suggests two natural categories of representations to consider: highest weight modules and nondegenerate Whittaker modules. The first family of modules, highest weight modules, has been studied extensively and has been shown to have a rich combinatorial structure. (See, for example, [Hum08].) The second family of modules, nondegenerate Whittaker modules, was introduced by Kostant in [Kos78] as an algebraic tool for understanding an analytic question about when an irreducible representation of a semisimple Lie group can be realized as a space of functions on the group. Kostant showed that this category of modules has an extremely simple structure. In [McD85], McDowell introduced a category N of modules that includes both nondegenerate Whittaker modules and highest weight modules. This is the category of Whittaker modules which is the main focus of my dissertation work. McDowell defined a class of standard Whittaker modules in this category which are parabolically induced from the irreducible nondegenerate Whittaker modules in Kostant s category. These standard Whittaker modules generalize the Verma modules of the highest weight category. He then showed that all irreducible Whittaker modules occur uniquely as quotients of these standard modules. Objects in N have finite length composition series, so in the Grothendieck group of the category, any object can be decomposed into a sum of its irreducible subquotients, counted with multiplicity. This led Miličić and Soergel [MS97] to ask the natural question: Question 1.1. What is the multiplicity of an irreducible Whittaker module in the composition series of a standard Whittaker module? In 1979, the analogous question about multiplicities of irreducible highest weight modules in the composition series of Verma modules was asked by Kazhdan and Lusztig in [KL79]. They conjectured that these multiplicities were determined by certain polynomials which relate two bases of the Hecke algebra of the associated Coxeter system. This conjecture did more than just establish multiplicity results: it implied that a deep combinatorial structure underlies the relationship between Verma modules and simple highest weight modules. The conjecture of Kazhdan and Lusztig was proven independently in 1981 by Beilinson and Bernstein [BB81], and Brylinski and Kashiwara [BK81] by establishing a connection between highest weight modules and perverse sheaves, using D-modules and the Riemann-Hilbert correspondence. The crucial ingredient of this proof was the decomposition theorem of algebraic geometry [BBD82], which provided the necessary semisimplicity in the category of perverse sheaves. The Whittaker modules in the category N correspond to D-modules with irregular singularities [MS14], so they cannot be compared to perverse sheaves using the Riemann-Hilbert correspondence. This means the perverse sheaf techniques used to solve the Kazhdan-Lusztig conjectures for Verma modules will not work for Whittaker modules, and any geometric realizations of the category must be in terms of D-modules. To answer Question 1.1 for Whittaker modules, Soergel and Miličić [MS97] and Backelin [Bac97] used a non-geometric approach, which answered the question of multiplicities but lacked the appealing combinatorial structure of Kazhdan-Lusztig polynomials. In 2011, Mochizuki proved a version of the decomposition theorem that holds for all holonomic D- modules, including those with irregular singularities [Moc11], providing the necessary framework for developing a D-module analogue to the Kazhdan-Lusztig algorithm of 1981 for the category N. The development of this algorithm for Whittaker modules with regular integral central character was the main accomplishment of my dissertation. The details of this construction are described in the following section. 2

3 1.2 Main Results The category of Whittaker modules consists of modules over a complex semisimple Lie algebra g that satisfy three finiteness conditions. Specifically, if h b is a Cartan subalgebra in a fixed Borel subalgebra of g with nilpotent radical n = [b, b], and Z(g) U(g) is the center of the universal enveloping algebra of g, then the category N of Whittaker modules contains all U(g)-modules which are finitely generated, Z(g)-finite, and U(n)-finite. For a choice of λ h and a Lie algebra morphism η n, McDowell constructed a standard Whittaker module M(λ, η), and showed that all irreducible Whittaker modules L(λ, η) appear uniquely as quotients of M(λ, η) [McD85]. When η = 0, the M(λ, 0) are Verma modules, and when η acts nontrivially on all root subspaces of g corresponding to simple roots (we say such η are nondegenerate), the M(λ, η) are the irreducible modules studied by Kostant in [Kos78]. So at these extreme values of η, the category has a very rigid (and well-studied) structure. In this project, I am primarily interested in what happens between these two extremes; that is, degenerate η which are not identically zero. Using the localization functor of Beilinson and Bernstein [BB81], the category N can be realized geometrically as the category M coh (D λ, N, η) of η-twisted Harish-Chandra sheaves, which are N- equivariant D λ -modules satisfying a compatibility condition determined by η. This category consists of holonomic D λ -modules, so its objects have finite length composition series and there is a welldefined duality in the category [BGK + 87]. The morphism η determines a subgroup W η of the Weyl group W of g, and from the parameters η n, C W η \W, and λ h, we construct a standard sheaf I(w C, λ, η), costandard sheaf M(w C, λ, η), and irreducible sheaf L(w C, λ, η). The relationship between the algebraic category N and the geometric category M coh (D λ, N, η) is given by the following theorem from my dissertation. Theorem 1.1. [Rom17] Let λ h be antidominant, η n a character, and C W η \W. Then Γ(X, M(w C, λ, η)) = M(w C λ, η). This relationship lets us formulate a geometric algorithm for computing the composition factors of a standard η-twisted Harish-Chandra sheaf and then translate that algorithm to the algebraic setting to answer Question 1.1. The statement of the algorithm is completely combinatorial (see [Rom17] for a precise statement), but the proof of the algorithm requires an appeal to the category M coh (D λ, N, η). The main result of my dissertation is the following theorem. Theorem 1.2. [Rom17] There is a canonical set of polynomials {P CD } C,D Wη\W generalizing the Kazhdan-Lusztig polynomials for the category N. These polynomials are constructed by pulling back irreducible η-twisted Harish-Chandra sheaves to certain Bruhat cells and computing the rank of the resulting D-module on the subvariety. After appropriate normalization, they can be recognized as the parabolic Kazhdan-Lusztig polynomials introduced by Deodhar in [Deo87]. This theorem answers Question 1.1 for the standard Whittaker module M( w C ρ, η) by developing an algorithm for calculating the multiplicities of its irreducible constituents. Here ρ is the half-sum of positive roots. The outline of the algorithm is as follows: Order elements of W η \W by the Bruhat order on longest coset representatives. Construct the matrix (P CD ( 1)) C,D Wη\W. This matrix is lower triangular and has 1 s on the diagonal. 3

4 Find the inverse matrix (µ CD ) C,D Wη\W. This matrix gives us our desired multiplicities. Corollary 1.3. [Rom17] The multiplicity of the irreducible Whittaker module L( w D ρ, η) in the standard Whittaker module M( w C ρ, η) is µ CD. By twisting by an invertible homogeneous O X -module, we can obtain an analogous result for standard modules parameterized by λ h which pair integrally and non-singularly with roots of g. (We say such modules have regular integral central character. ) This algorithm establishes the underlying combinatorial structure of the category N and gives deeper meaning to the multiplicities calculated by Miličić and Soergel in [MS97]. In addition, it provides a more efficient method for computing these multiplicities without appealing to the Kazhdan-Lusztig algorithm for Verma modules as Miličić and Soergel did. 1.3 Future Directions In the work described above, I have developed tools for analyzing the category N both algebraically and geometrically. This leaves me well-equipped to address other natural questions about this category. The most obvious of these questions is what happens in the case of singular and nonintegral central character. Question 1.2. Can the algorithm from my dissertation be extended to address multiplicity questions for standard Whittaker modules with arbitrary central character? For Verma modules, the Kazhdan-Lusztig algorithm can be extended to arbitrary λ h by reducing the algorithm to the integral Weyl group, which is constructed from the roots with which λ pairs integrally. (See, for example, [Soe90, KT00].) However, the Whittaker setting requires a subtler approach which relates the sub-weyl group determined by η and the integral Weyl group. I am currently developing techniques for generalizing the Whittaker algorithm that combine the strategy in the Verma module setting with explicit computations from [Rom17]. By using translation functors and Theorem 1.1, one can establish combinatorial conditions that dictate when the global sections of irreducible η-twisted Harish-Chandra sheaves with singular central character vanish, and this can be used to compute multiplicities for singular λ. To address the non-integral case, a parabolic set of roots can be constructed by taking the union of the root system determined by η and the positive roots. When intersected with the integral root system, this yields a smaller parabolic set of roots, which determines a subset of simple roots depending on both η and λ. I conjecture that the Kazhdan-Lusztig polynomials for non-integral Whittaker modules are the polynomials corresponding to this subset. The conditions on irreducibility of standard Whittaker modules established in [Luk04] support this conjecture, and I am working to generalize the techniques used in non-integral Verma module algorithm to prove it. An interesting consequence of the algorithm in [Rom17] is its relationship to the Kazhdan- Lusztig algorithm for generalized Verma modules in [Mil]. The combinatorics arising in these algorithms displays a duality that leads to the following question. Question 1.3. Is the combinatorial duality present in the Kazhdan-Lusztig polynomials for generalized Verma modules and the Kazhdan-Lusztig polynomials for Whittaker modules a shadow of a deeper duality between these two categories? The equivalence between blocks of N and singular blocks of category O established in [MS97] and the parabolic-singular duality between singular blocks of O and regular blocks of parabolic category O p described in [BGS96] indicate that a duality between Whittaker modules and generalized Verma modules should exist, but this relationship has not yet been made precise in the 4

5 literature. A first step in formalizing this idea is to examine the role of projective objects in N in an attempt to realize blocks of N as module categories over the endomorphism ring of some projective generator, following the approach of Soergel in [BGS96]. This introduces a Koszul ring, and we can calculate the corresponding Koszul dual ring, with the final goal of realizing blocks of parabolic category O p as modules over the Koszul dual ring. The algebraic structure of N established in [Rom17] provides a solid foundation with which I can approach this problem. In addition to these questions, it is natural to ask whether other well-established results of the category of highest weight modules extend to N. In [Jan79], Jantzen introduced a canonical filtration of Verma modules which provided a beautiful conceptual proof of BGG reciprocity for highest weight modules, and he conjectured that this filtration is compatible in a natural way with embeddings of Verma modules. This became known as the Jantzen Conjecture. It was discovered by Gabber and Joseph [GJ81] that this conjecture establishes detailed information about the coefficients of the Kazhdan-Lusztig polynomials for Verma modules, and a proof of it implies the Kazhdan-Lusztig conjectures. The standard Whittaker modules in N have similar structural properties to Verma modules, so it is reasonable to consider the following problem. Question 1.4. Can one define Jantzen filtrations in N in order to develop and prove a conjecture analogous to the Jantzen conjecture for N? In [BB93], Beilinson and Bernstein provide a proof of the Jantzen conjectures using weight filtrations on the corresponding perverse sheaves. However, as mentioned previously, the irregular singularities of modules in M coh (D λ, N, η) make methods of perverse sheaves intractable for N. Therefore, the geometric development of Jantzen filtrations for Whittaker modules would require detailed analysis of holonomic D-modules with irregular singularities, building on the structure established in [Moc11]. In this setting, the interactions between Hodge theory and representation theory have not yet been thoroughly studied, and there are many possibilities for future development. The foundational results on such D-modules established in [Rom17] provide an excellent launching point to explore this new territory. Furthermore, there is an alternate algebraic approach to this problem using Soergel bimodules, which is described below. The category of Whittaker modules could also be approached using a different set of tools. Inspired by the celebrated algebraic proof of the Kazhdan-Lusztig conjectures by Elias-Williamson in [EW14], one could examine category N using Soergel bimodules with the goal of providing a purely algebraic proof of Corollary 1.3 that doesn t appeal to geometry. Indeed, Theorem 1.2 can be reformulated in terms of the anti-spherical module for the Hecke algebra associated to W [Soe97]. In [LW17], Libedinsky-Williamson use a diagrammatic category of Soergel bimodules (which they refer to as the anti-spherical category ) to categorify the anti-spherical module, and this categorification establishes positivity of coefficients of parabolic Kazhdan-Lusztig polynomials. This leads to the following question. Question 1.5. Can the anti-spherical category of Libedinsky-Williamson be used to provide a purely algebraic proof of Corollary 1.3? In [RW15], Riche-Williamson relate the anti-spherical category of the affine Weyl group to representations of algebraic groups, establishing the importance of this category in modular representation theory. Therefore, an answer to Question 1.5 would also illuminate the role of Whittaker modules in modular representation theory. Additionally, this approach could present an alternate avenue to Question 1.4 by building on Williamson s recent algebraic proof of the Jantzen Conjecture for Verma modules using Soergel bimodules [Wil16]. 5

6 2 An orbit model for nilpotent Gelfand pairs 2.1 Motivation In addition to my main dissertation work, I have an ongoing collaborative project studying nilpotent Gelfand pairs. A Gelfand pair (G, K) consists of a locally compact topological group and a compact subgroup K G such that the algebra L 1 (K\G/K) of integrable K-invariant functions on G is commutative under convolution. Gelfand pairs and their corresponding homogeneous spaces G/K generalize the notion of a Riemannian symmetric space, and they arise naturally in harmonic analysis and representation theory of Lie groups. This project uses Gelfand pairs to study the representation theory of solvable Lie groups. When considering Gelfand pairs for solvable Lie groups G, one quickly runs into some problems. For example, if G is simply connected, there may be no non-trivial compact subgroups. To remedy this, one can consider pairs (K G, K), where K is a compact subgroup of the automorphism group of G, and G is a solvable Lie group. In [BJR90], Benson, Jenkins and Ratcliff showed that the classification of Gelfand pairs of this form reduces to the classification of nilpotent Gelfand pairs. These are pairs (K, N) of a connected and simply connected nilpotent Lie group N and a compact subgroup K of automorphisms of N such that the algebra L 1 K (N) is an abelian algebra under convolution. The nilpotent groups N arising in this way are necessarily 2-step, so much of their structure and representation theory is related to that of the Heisenberg groups. This project develops a topological model for the spectra of a certain class of nilpotent Gelfand pairs. 2.2 Main Results The Gelfand space (or spectrum) of a nilpotent Gelfand pair (K, N) is the set of continuous non-zero algebra homomorphisms from L 1 K (N) to C. This can be identified, via integration, with the space (K, N) of bounded K-spherical functions on N. A bounded K-spherical function is a smooth bounded K-invariant function φ : N C that is a simultaneous eigenfunction for all K-invariant differential operators on N. The space (K, N) can in turn be identified with a certain class of unitary representations of G = K N. The K-spherical representations of G are Ĝ K = {ρ Ĝ ρ has a 1-dimensional space of K-fixed vectors}. Here Ĝ is the unitary dual of G. Given a K-spherical representation, there is a corresponding bounded K-spherical function which is formed by taking the diagonal matrix coefficient for a K- fixed vector of unit length. In the last twenty years, some interesting topological models for the Gelfand space of a nilpotent Gelfand pair have emerged in the literature. These topological models illustrate the beautiful geometric structure of the moduli space ĜK. One such model was proposed by Benson and Ratcliff in [BR08]. In this paper, they establish a bijection Ψ between the Gelfand space of a nilpotent Gelfand pair and a set A(K, N) of K-spherical orbits in the dual of the Lie algebra of N. The motivation behind this construction is the Orbit Method philosophy of representation theory which asserts that irreducible unitary representations of a Lie group should correspond to coadjoint orbits in the dual of its Lie algebra. In [BR08], Benson and Ratcliff also show that Ψ is a homeomorphism for three classes of nilpotent Gelfand pairs: (K, N) with N abelian, (K, N) = (U(n), H n ) with H n a Heisenberg group, and (K, N) = (O(d), F d ) with F d the free 2-step nilpotent group. Additionally, they propose the following conjecture. Conjecture 2.1. [BR08] The map Ψ : (K, N) A(K, N) is an isomorphism for all nilpotent Gelfand pairs (K, N). 6

7 This conjecture was proven for nilpotent Gelfand pairs (K, H n ) where K U(n) in [BR13, BR15a, BR15b]. In joint work with H. Friedlander, W. Grodzicki, W. Johnson, G. Ratcliff, B Strasser, and B. Wessel, we have proven Conjecture 2.1 for a broader class of nilpotent Gelfand pairs. Our precise result is described below. For a nilpotent Gelfand pair (K, N), let n = Lie N and write n = V z, where z is the center of n and [V, V] z. We can endow n with an inner product, such that V z. Our main result in [FGJ + 17] is the following theorem. Theorem 2.1 (Friedlander, Grodzicki, Johnson, Ratcliff, Romanov, Strasser, Wessel). Let (K, N) be a nilpotent Gelfand pair satisfying the following two conditions: (i) Generic orbits of the restricted action of K on the center z are of codimension one; and (ii) For any fixed unit base point A z, the skew-symmetric form (X, Y ) [X, Y ], A on V is nondegenerate. Then the map Ψ associating a K-spherical function to the associated K-spherical orbit in n is a homeomorphism. In contrast to the case-by-case approach which has traditionally been used in the study of nilpotent Gelfand pairs, our proof of Theorem 2.1 uses a general argument that applies to several infinite families of nilpotent Gelfand pairs. 2.3 Future Directions There is also a notion of Gelfand pairs in the finite group setting. The combinatorial nature of this subject makes it fertile ground for undergraduate research projects, and I intend to take my work in Gelfand pairs in this direction with the goal of developing a secondary research program that can include undergraduate researchers. For a finite group G with subgroup K, a pair (G, K) is a Gelfand pair if the algebra L(K\G/K) of K-bi-invariant functions on G is commutative. One way that finite Gelfand pairs arise is through the study of parking functions, which are algebro-combinatorial objects related to hashing functions, labeled trees, hyperplane arrangements, and non-crossing setpartitions. For a finite group Γ, the symmetric group S n acts on the product Γ n = Π n i=1 Γ by permuting the indices. This can be interpreted as a parking function. If G n = Γ n S n is the wreath product formed from this action, then the group K n = n S n is a subgroup of G n, where n is the diagonal in Γ n. Aker and Can showed in [AC12] that if Γ is abelian, (G n, K n ) is a Gelfand pair. In this paper they also conjectured that for Γ non-abelian, there is some large enough n such that (G n, K n ) is not a Gelfand pair. In [BR], Benson and Ratcliff set out to find these cracking points suggested in [AC12] where (G n, K n ) fails to be a Gelfand pair. They proved that for Γ non-abelian, there is a non-negative integer N(Γ) such that for all m N(Γ), (G m, K m ) is not a Gelfand pair, and showed that for dihedral groups Γ = D p for p an odd prime, N(Γ) = 6, regardless of p. These results led to some natural questions about this class of Gelfand pairs: Question 2.1. Can these cracking points be arbitrarily large? That is, for every positive integer n, is there some group Γ such that N(Γ) = n? Question 2.2. What is the relationship between N(Γ) and the representation theory of Γ? For example, is there a relationship between the maximal dimension of irreducible representations of Γ and N(Γ)? 7

8 Answering these questions would shed more light on the theory of finite Gelfand pairs and their relationship to the representation theory of the underlying groups. Furthermore, the combinatorial nature of the questions makes them particularly accessible to undergraduate students with experience in basic group theory. Using the techniques developed in [BR] to tackle these questions would be an excellent launching place for a secondary research program. References [AC12] [Bac97] [BB81] K. Aker and M. B. Can. From parking functions to Gelfand pairs. Proc. Amer. Math. Soc., 140(4): , E. Backelin. Representation of the category O in Whittaker categories. Internat. Math. Res. Notices, (4): , A. Beĭlinson and J. Bernstein. Localisation de g-modules. C. R. Acad. Sci. Paris Sér. I Math., 292(1):15 18, [BB93] A. Beĭlinson and J. Bernstein. A proof of Jantzen conjectures. In I. M. Gel fand Seminar, volume 16 of Adv. Soviet Math., pages Amer. Math. Soc., Providence, RI, [BBD82] A. A. Beĭlinson, J. Bernstein, and P. Deligne. Faisceaux pervers. In Analysis and topology on singular spaces, I (Luminy, 1981), volume 100 of Astérisque, pages Soc. Math. France, Paris, [BGK + 87] A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers. Algebraic D-modules, volume 2 of Perspectives in Mathematics. Academic Press, Inc., Boston, MA, [BGS96] [BJR90] [BK81] [Blo81] A. Beilinson, V. Ginzburg, and W. Soergel. Koszul duality patterns in representation theory. J. Amer. Math. Soc., 9(2): , C. Benson, J. Jenkins, and G. Ratcliff. On Gel fand pairs associated with solvable Lie groups. Trans. Amer. Math. Soc., 321(1):85 116, J.-L. Brylinski and M. Kashiwara. Kazhdan-Lusztig conjecture and holonomic systems. Invent. Math., 64(3): , R. E. Block. The irreducible representations of the Lie algebra sl(2) and of the Weyl algebra. Adv. in Math., 39(1):69 110, [BR] C. Benson and G. Ratcliff. A family of finite Gelfand pairs associated with wreath products. unpublished manuscript. [BR08] [BR13] [BR15a] C. Benson and G. Ratcliff. The space of bounded spherical functions on the free 2-step nilpotent Lie group. Transform. Groups, 13(2): , C. Benson and G. Ratcliff. Geometric models for the spectra of certain Gelfand pairs associated with Heisenberg groups. Ann. Mat. Pura Appl. (4), 192(4): , C. Benson and G. Ratcliff. Spaces of bounded spherical functions on Heisenberg groups: part I. Ann. Mat. Pura Appl. (4), 194(2): ,

9 [BR15b] C. Benson and G. Ratcliff. Spaces of bounded spherical functions on Heisenberg groups: part II. Ann. Mat. Pura Appl. (4), 194(2): , [Deo87] V. V. Deodhar. On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials. J. Algebra, 111(2): , [EW14] [FGJ + 17] [GJ81] B. Elias and G. Williamson. The Hodge theory of Soergel bimodules. Ann. of Math. (2), 180(3): , H. Friedlander, W. Grodzicki, W. Johnson, G. Ratcliff, A. Romanov, B. Strasser, and B. Wessel. An orbit model for the spectra of a class of nilpotent Gelfand pairs. In preparation, O. Gabber and A. Joseph. Towards the Kazhdan-Lusztig conjecture. Ann. Sci. École Norm. Sup. (4), 14(3): , [Hum08] J. E. Humphreys. Representations of semisimple Lie algebras in the BGG category O, volume 94 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, [Jan79] [KL79] J. C. Jantzen. Moduln mit einem höchsten Gewicht, volume 750 of Lecture Notes in Mathematics. Springer, Berlin, D. Kazhdan and G. Lusztig. Representations of Coxeter groups and Hecke algebras. Invent. Math., 53(2): , [Kos78] B. Kostant. On Whittaker vectors and representation theory. Invent. Math., 48: , [KT00] M. Kashiwara and T. Tanisaki. Characters of irreducible modules with non-critical highest weights over affine Lie algebras. In Representations and quantizations (Shanghai, 1998), pages China High. Educ. Press, Beijing, [Luk04] D. Lukic. Twisted Harish-Chandra sheaves and Whittaker modules. PhD thesis, University of Utah, [LW17] N. Libedinsky and G. Williamson. The anti-spherical category, preprint arxiv: [McD85] E. McDowell. On modules induced from Whittaker modules. J. Algebra, 96(1): , [Mil] [Moc11] [MS97] [MS14] D. Miličić. Localization and representation theory of reductive Lie groups. unpublished manuscript available at T. Mochizuki. Wild harmonic bundles and wild pure twistor D-modules. Astérisque, (340), D. Miličić and W. Soergel. The composition series of modules induced from Whittaker modules. Comment. Math. Helv., 72(4): , D. Miličić and W. Soergel. Twisted Harish-Chandra sheaves and Whittaker modules: The nondegenerate case. Developments and Retrospectives in Lie Theory: Geometric and Analytic Methods, 37: ,

10 [Rom17] [RW15] [Soe90] [Soe97] A. Romanov. A Kazhdan-Lusztig algorithm for Whittaker modules. PhD thesis, University of Utah, S. Riche and G. Williamson. Tilting modules and the p-canonical basis, preprint arxiv: W. Soergel. Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe. J. Amer. Math. Soc., 3(2): , W. Soergel. Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules. Represent. Theory, 1:83 114, [Wil16] G. Williamson. Local Hodge theory of Soergel bimodules. Acta Math., 217(2): ,