Research Statement

Jay Taylor

Groups form the abstract framework for studying the natural notion of symmetry and are ubiquitous in mathematics. The building blocks for constructing all groups are the simple groups. Often one hopes to reduce a question about groups to statements involving only groups that are close to being simple. A key example of this is given by Isaacs–Malle–Navarro’s reduction of the McKay Conjecture [IMN07]. Such reductions make it essential to have a detailed understanding of simple groups. The celebrated Classification of Finite Simple Groups states that the vast majority of finite simple groups arise as central quotients of finite reductive groups. Each finite reductive group G(q) is defined with respect to a finite field Fq of q elements. The principal examples are the matrix groups GLn(q), SLn(q), and Sp2n(q), whose matrices have entries in the field Fq. It is known that the quotients PSLn(q) and PSp2n(q) of SLn(q) and Sp2n(q) by their centers are generically simple. My research concerns the character theory of finite reductive groups, which is a fundamental tool for studying finite groups. A character of a finite group G is a function G → C defined by taking the trace of a matrix obtained from a homomorphism G → GLn(C). The characters of a group encode an impressive amount of information and, somewhat surprisingly, are often computable even without knowing the underlying homomorphisms G → GLn(C) from which they are defined. For these reasons, characters have become an indispensable tool for studying finite groups. The basic object of study in character theory is the set of irreducible characters Irr(G). The overarching theme of my research concerns the values of irreducible characters of finite reductive groups at unipotent elements. The study of all character values can largely be reduced to the case of values at unipotent elements, which makes this an important case to consider. Moreover, it is this case where the connections to geometry are most rich, such as through the Springer correspondence and studying singularities in the closures of unipotent conjugacy classes. An important viewpoint on character values at unipotent elements comes from Kawanaka’s generalized Gelfand–Graev characters (GGGCs), which were introduced in [Kaw85; Kaw86]. These are a family of characters of G that are defined whenever the characteristic p of the field Fq is good (p > 5 is sufficient). In the last few years these characters have seen remarkable applications to the following fundamental questions:

• determining the action of automorphisms of G on the set Irr(G) of irreducible characters, by work of Cabanes–Späth [CS17] and myself [Tay18a], • bounding the values of irreducible characters, by work of Bezrukavnikov–Liebeck–Shalev–Tiep [BLST17] and my recent joint work with Tiep [TT18], • and determining the `-modular decomposition matrices where ` 6= p is a prime, by work of Dudas–Malle [DM16; DM18] and my current ongoing joint work with Brunat and Dudas [BDT],

This last application encodes how homomorphisms G → GLn(C) defined over C are related to homomorphisms G → GLn(F`) defined over an algebraically closed field of characteristic ` > 0. All of the above applications rely on our understanding of the decomposition of a GGGC into its irreducible constituents. When p is a good prime, I established the best general statements regarding such a decomposition in [Tay16]. This generalizes previous work of Lusztig [Lus92] whose results were

page 1 Jay Taylor Research Statement obtained under the assumption that the characteristic p > 0 is sufficiently large. Such assumptions on the characteristic are incredibly prohibitive when invoking the Classification of Finite Simple Groups. In general, it is difficult to say whether an irreducible character will occur as a constituent of a GGGC. Answering this question is an ongoing focus of my research. I have developed a strategy for solving this problem which involves:

• relating the parameterization of the irreducible characters Irr(G) via Harish-Chandra theory to the labelling given by Lusztig [Lus84], • computing the decomposition of the Deligne–Lusztig induction map from a Levi subgroup, • evaluating the characteristic function of a character sheaf at unipotent elements, • determining the decomposition of an irreducible character as a linear combination of characteristic functions of character sheaves.

In the following sections I will provide some background on ﬁnite reductive groups and elaborate on the above projects and future work in more detail.

Background A finite reductive group is defined to be the fixed points GF of a connected reductive algebraic group, defined over an algebraic closure F = Fq of the finite field Fq, under a Frobenius endomorphism F : G → G corresponding to the finite field Fq. The main example is the case where G = GLn(F) q F and F (xij ) = (xij ) for all (xij ) ∈ G. We then have G is the finite general linear group GLn(q). By realizing the finite group GF as a subgroup of the algebraic group G we may use the geometry of the group G when studying GF . The Frobenius endomorphism F is the key to this relationship. A particularly convenient setting for studying the irreducible characters Irr(GF ) is to view them as F a basis for the C-vector space Class(G ) of all class functions, i.e., functions that are constant on conjugacy classes. In his seminal work [Lus85] Lusztig has introduced his geometric theory of character sheaves, which provides the theoretical framework for studying the values of irreducible characters. Each such character sheaf is a G-equivariant simple perverse sheaf on the algebraic group G. From the theory of character sheaves one obtains a new basis C(GF ) ⊆ Class(GF ) for the space of class functions. It has been conjectured by Lusztig that this basis C(GF ) is very close to the basis of F irreducible characters. For instance, if G = GLn(q) then these functions are precisely the irreducible characters. Lusztig’s conjecture has a positive solution in many cases but remains open in general. Even when the conjecture is known to hold the entries of the change of basis matrix relating Irr(GF ) and C(GF ) are not known explicitly.

Evaluating Characteristic Functions of Character Sheaves

Current and Previous Work

For each irreducible character χ ∈ Irr(GF ) and characteristic function X ∈ C(GF ) there exists a scalar c(χ, X ) ∈ C such that X χ = c(χ, X )X . X ∈C(GF )

page 2 Jay Taylor Research Statement

Computing the value of an irreducible character χ ∈ Irr(GF ) at an element g ∈ GF can be viewed as the following two step process:

1) for any function X ∈ C(GF ) compute the value X (g), 2) for any X ∈ C(GF ) with X (g) 6= 0 compute the scalar c(χ, X ).

That the values of X ∈ C(G) can be computed at all comes from the fact that X is obtained from a geometric object, namely a perverse sheaf. In this section I will describe some results concerning step 1 of this process when g ∈ GF is a unipotent element. If O ⊆ G is an F -stable unipotent conjugacy class of G then OF = O ∩ GF is a union of GF -conjugacy classes. To determine X (g) explicitly involves choosing one of the GF - conjugacy classes contained in OF that satisﬁes desirable properties. In many, but not all, cases Shoji has made a choice of such a class whose elements are called split unipotent elements [Sho87]. Even though a theory of split unipotent elements does not exist in general, I was able to establish the following.

Theorem 1 (Taylor, [Tay14]). Assume a theory of split unipotent elements exists for GF then for each function X ∈ C(GF ) and unipotent element u ∈ GF the value X (u) is explicitly computable via a recursive algorithm. Moreover, a theory of split unipotent elements exists if p is a good prime for G and Z(G) is connected.

Future Work

If p is a good prime for G then a theory of split unipotent elements exists for most groups, even without the assumption that Z(G) is connected. The most prominent case where this theory is missing is when G = Spinn(F) is a spin group. For such a group there exists a surjective homomorphism Spinn(F) → SOn(F) onto a special orthogonal group with kernel of size 2. In other words, Spinn(F) is a double cover of SOn(F). In [Sho07] Shoji explicitly described split unipotent elements for the special orthogonal group

SOn(F). The surjective homomorphism Spinn(F) → SOn(F) restricts to a bijection between the unipotent elements of Spinn(F) and those of SOn(F). By using this natural relationship and the tools developed by Shoji I aim to tackle the following problem.

Problem 2. Using the existence of split unipotent elements in special orthogonal groups develop a theory of split unipotent elements for spin groups.

This problem constitutes the last major stumbling block in developing a theory of split unipotent elements when p is a good prime for G. As mentioned in the introduction, studying character values at arbitrary elements can often be reduced to the case of unipotent elements. Speciﬁcally one studies the ◦ F F values of the irreducible characters Irr(CG(s) ) at unipotent elements, where s ∈ G is a semisimple ◦ element and CG(s) is its connected centralizer. When considering exceptional groups such as E8 one encounters low dimensional spin groups when ◦ studying the connected centralizers CG(s) of semisimple elements. Hence, even partial results for Problem 2 would have applications for determining character tables of exceptional groups at non- unipotent elements. When trying to make eﬀective use of these reduction arguments to unipotent elements one comes across the following problem.

page 3 Jay Taylor Research Statement

Problem 3. Generalize the result of Theorem 1 to arbitrary elements by reducing to the case of ◦ F unipotent elements in the connected centralizer CG(s) of a semisimple element s ∈ G .

Obtaining a solution to this problem is essential for determining the values of irreducible characters at non-unipotent elements. I believe this project would be suitable for a graduate student.

Decomposing Generalized Gelfand–Graev Characters

Current and Previous Work

Let us assume that p is a good prime for G so that Kawanaka’s generalized Gelfand–Graev characters (GGGCs) are deﬁned. A major leap forward in the study of GGGCs was given by Lusztig in [Lus92]. In particular, assuming p > 0 is suﬃciently large Lusztig gave a formula decomposing the GGGCs as an explicit linear combination of characteristic functions of intersection cohomology complexes, which are supported on the closure of an F -stable unipotent conjugacy class of G. The functions obtained in this way, from intersection cohomology complexes, are very closely related to the basis C(GF ) ⊆ Class(GF ) of the space of class functions obtained from Lusztig’s character sheaves. In [Tay16] I was able to weaken the assumption on p in Lusztig’s work to the case of acceptable primes. In most cases a prime is acceptable for G if and only if it is good for G.

Theorem 4 (Taylor, [Tay16]). If p is an acceptable prime for G then Lusztig’s formula for GGGCs remains valid.

Assuming p is suﬃciently large Lusztig managed to use this formula to prove the following amazing relationship between the irreducible characters of GF and the unipotent conjugacy classes of G. He F ∗ showed that for any χ ∈ Irr(G ) there exists a unique F -stable unipotent conjugacy class Oχ of G, called the wave-front set of χ, which satisﬁes the following properties:

∗ F • χ is a constituent of some GGGC Γu with u ∈ Oχ , G ∗ G • if χ occurs as a constituent of a GGGC Γv then dim v 6 dim Oχ, where v is the G-conjugacy class containing v.

∗ Roughly speaking Oχ is the largest class whose corresponding GGGCs see the irreducible character χ. The existence of such a class had been previously conjectured by Kawanaka [Kaw85]. The following settles Kawanaka’s conjecture in full generality.

∗ Theorem 5 (Taylor, [Tay16]). If p is a good prime for G then the wave front set Oχ exists for any irreducible character χ ∈ Irr(GF ).

One nice consequence of the results in [Tay16] is that the following result of Geck [Gec94], originally proved under the assumption that p is sufficiently large, is now known to hold under significantly milder restrictions on p. Namely, if G is a group of classical type and p > 2 is an odd prime then the 2- modular decomposition matrix of GF has a lower unitriangular shape. In particular, we have a canonical labelling of the 2-modular irreducible Brauer characters of GF . For odd primes ` 6= p it is still an open and difficult problem to determine a natural labelling for the irreducible `-Brauer characters of GF . This is the focus of my ongoing joint work with Brunat and Dudas [BDT].

page 4 Jay Taylor Research Statement

Future Work

As mentioned in the introduction, a major focus of my current research aims to determine the multiplicities of irreducible characters in GGGCs. In general these multiplicities are very complicated but it is known by [Lus92; Tay16] that the multiplicity of an irreducible character χ ∈ Irr(GF ) in ∗ F a GGGC Γu is “small”, in a strong sense, whenever u ∈ Oχ is contained in the wave-front set. In [DLM14] Digne–Lehrer–Michel obtained a precise formula for these multiplicities under very restrictive assumptions on χ. A solution to the following problem should generalize and build on the results of Digne–Lehrer–Michel.

Problem 6. Assuming that Z(G) is connected compute explicitly the multiplicity of χ in a GGGC Γu ∗ F F whenever u ∈ Oχ is contained in the wave front set of χ ∈ Irr(G ).

In [Tay13] I made progress on solving one of the last open conjectures of Kawanaka concerning GGGCs. Namely, that the GGGCs form a basis for the Z-module of unipotently supported virtual characters of GF . As remarked in [Tay13], the main problem for proving this conjecture in full stems from not having a solution to Problem 6 for spin groups. Were a solution to Problem 6 attained then this conjecture could ﬁnally be settled. Previous work on this conjecture formed the basis of Dudas– Malle’s recent result [DM18] concerning the modular irreducibility of cuspidal unipotent characters.

Action of Automorphisms

Current and Previous Work

One of the basic situations one considers in finite group theory is that of a finite group G with normal subgroup N C G. Understanding the relationship between N and G is crucial to reducing a question about G to finite simple groups. The group G acts on N via automorphisms, which are induced by conjugation. In turn, we obtain an action of G on the set of irreducible characters Irr(N). Clifford’s Theorem gives a relationship between the irreducible characters Irr(G) and Irr(N) with the proviso that one understands how G acts on Irr(N). Instead of considering this specific situation one simply aims to describe the action of the automorphism group Aut(G) of a finite group G on the set of irreducible characters Irr(G) given by χ 7→ σχ = χ ◦ σ−1, where χ ∈ Irr(G) and σ ∈ Aut(G). This question has recently come to promi- nence due to the reduction of the McKay Conjecture to finite simple groups [IMN07] and several other reductions that have appeared in its wake. It is currently a major focus of research to understand this action, with the most important open case being when G = GF is a finite reductive group. F In the case where G = SLn(q) this action was described by Cabanes–Späth [CS17] using Kawanaka’s solution [Kaw85] to Problem 6 for GLn(q). Using a weak solution to Problem 6, together with techniques coming from Harish-Chandra theory, I was able to achieve the following result regarding automorphisms.

F Theorem 7 (Taylor, [Tay18a]). Assume G = Sp2n(q) is a symplectic group with q odd. Given any irreducible character χ ∈ Irr(GF ) and automorphism σ ∈ Aut(GF ) the character σχ can be explicitly described.

page 5 Jay Taylor Research Statement

Future Work

There is another group that acts naturally on the irreducible characters of GF . For any χ ∈ Irr(GF ) F and g ∈ G it is known that χ(g) ∈ Q is contained in an algebraic closure of the rationals. Moreover, F the absolute Galois group Gal(Q/Q) acts on Irr(G ) by simply applying the automorphism to the values of the character. Understanding this action is related to understanding the smallest subﬁeld of Q which contains the values of χ. For instance, if χ is ﬁxed under every such Galois automorphism then we know that F χ(g) ∈ Q is rational for each g ∈ G . Aside from this, the action of Gal(Q/Q) forms a fundamental component in understanding Navarro’s Galois–McKay Conjecture [Nav04], of which a reduction to simple groups has recently been announced by Navarro–Späth–Vallejo. A natural generalization of Theorem 7 is to consider the following.

F Problem 8. Assuming G = Sp2n(q) is a ﬁnite symplectic group, describe the action of the absolute F Galois group Gal(Q/Q) on Irr(G ).

I am currently working on this project in collaboration with A. A. Schaeﬀer Fry. This builds on our previous joint work [SFT18a; SFT18b] where we studied the action of Gal(Q/Q) on groups such as SLn(q) and SUn(q).

Bounding Character Values Current and Previous Work

It is an incredibly diﬃcult problem to determine the value χ(g) of an irreducible character χ ∈ Irr(GF ) at an element g ∈ GF . However, one elementary statement that can be made using character theory is that |χ(g)| 6 χ(1) or, equivalently, that |χ(g)|/χ(1) 6 1. The term |χ(g)|/χ(1) is called a character ratio and bounding such terms has important applications in ﬁnite group theory. For instance, a result going back to Frobenius states that an element g ∈ GF is a commutator if and only if

X χ(g) 6= 0. χ(1) χ∈Irr(GF )

This fact, together with bounds on character ratios, formed the foundation of Liebeck–Shalev– O’Brien–Tiep’s proof [LOST10] of Ore’s Conjecture, which states that every element in a ﬁnite simple group is a commutator. α One useful style of bound, called an exponential bound, is of the form |χ(g)| 6 χ(1) g , where F 0 6 αg 6 1 is a constant depending on the element g ∈ G . The important point here, as illustrated by the above sum, is that the bound should hold for all irreducible characters χ ∈ Irr(GF ). The group G has a class of subgroups known as Levi subgroups. These subgroups are important as they are again connected reductive algebraic groups. For instance, if G = GLn(F) then the Levi subgroups of G are isomorphic to GLm1 (F) × · · · × GLmk (F) with m1 + ··· + mk = n. If L 6 G is an F -stable Levi subgroup of G then following Bezrukavnikov–Liebeck–Shalev–Tiep [BLST17] we deﬁne

αG(L) to be 0 if L is a torus and dim uL αG(L) = max G 16=u∈LF dim u unipotent

page 6 Jay Taylor Research Statement otherwise. Here the maximum is taken over all non-identity unipotent elements of L and uL and uG denote the L-conjugacy class and G-conjugacy class, respectively, containing u. In [BLST17] the authors obtain a bound on |χ(g)|, in terms of χ(1)αG(L), assuming that g ∈ GF is an element whose centralizer CG(g) 6 L is contained in a split Levi subgroup L 6 G, i.e., we have L is the Levi complement of an F -stable parabolic subgroup of G. It is noted in [BLST17] that the exponent αG(L) is ultimately best possible in this generality. The following considerably generalizes the main result in [BLST17] by removing the assumption that L is split when Z(G) is connected. Its proof relies on many deep results relating irreducible characters and character sheaves as well as Kawanaka’s GGGCs and Theorem 5.

Theorem 9 (Taylor–Tiep, [TT18]). There exists a function f : N → N such that the following statement holds. Assume that G is a connected reductive algebraic group of semisimple rank r with connected centre, F : G → G is a Frobenius endomorphism, and p is a good prime for G. Then for F ◦ any F -stable Levi subgroup L 6 G and any element g ∈ L with connected centralizer CG(g) 6 L we have that α (L) |χ(g)| 6 f (r) · χ(1) G for any irreducible character χ ∈ Irr(GF ).

Future Work

F ◦ If u ∈ G is a unipotent element then the above theorem does not apply because CG(u) will never be contained in a proper Levi subgroup. Hence, this case must be considered using diﬀerent methods. In general, the value χ(u) of an irreducible character χ ∈ Irr(GF ) at a unipotent element u ∈ GF will be a polynomial in q. The following would give some asymptotic bound on χ(u).

Problem 10. Assume p is a good prime for G, χ ∈ Irr(GF ) is an irreducible character, and u ∈ GF is a unipotent element. Determine the degree of χ(u) as a polynomial in q.

∗ Using known results about GGGCs and the the wave-front set Oχ one can conjecture that the ∗ G F degree of this polynomial is (dim Oχ−dim u )/2. When u = 1 ∈ G is the identity then this conjecture is known to be correct. One way to approach this problem would be to ﬁrst investigate the values X (u) with X ∈ C(GF ) the characteristic function of a character sheaf. Here the result of Theorem 1 will be useful.

Computational Projects Root Data

To each connected reductive algebraic group G one may associate a combinatorial object R(G) = (X,Y ) known as its root datum. This consists of a pair of r × n integer matrices whose product XY t is a Cartan matrix, where Y t is the transpose of Y . There is a natural equivalence relation on such pairs given by (X,Y ) ∼ (X0,Y 0) if (X0,Y 0) = (XA−1,YAt ) for some invertible integer matrix 0 A ∈ GLn(Z). Moreover, we have two connected reductive algebraic groups G and G are isomorphic if and only if R(G) ∼ R(G0). A natural question that arises is the following: can one give an eﬀective parameterization of the equivalence classes of root data? Recently I answered this question by obtaining a combinatorial

page 7 Jay Taylor Research Statement parameterization of these orbits [Tay17]. In current joint work with Jean Michel we have developed an algorithm, implemented in the GAP3 package CHEVIE [Mic15], which enables one to concretely compute this label so that one has an eﬀective way of detecting when two connected reductive algebraic groups are isomorphic.

Future Work

A Weyl group W is a subgroup of the orthogonal group On(R) 6 GLn(R) which is generated by reflections. Moreover, a subgroup of W generated by reflections is called a reflection subgroup. If P 6 W is a reflection subgroup and ψ ∈ Irr(P ) is an irreducible character then one may obtain a W character IndP (ψ) of W known as the induced character. A natural question that arises when working with finite reductive groups is to determine the multiplicity of an irreducible character χ ∈ Irr(W ) in W the induced character IndP (ψ). The natural case is when W = Sn is the symmetric group and P = Sa × Sb is a Young subgroup with a + b = n. In this case the multiplicities are given by the classical Littlewood–Richardson coefficients. I was able to show the following analogue of this fact for type Dn Weyl groups.

Theorem 11 (Taylor, [Tay15]). If W is a Weyl group of type Dn and P is a reﬂection subgroup of W type Da × Db with a + b = n then the multiplicity of χ ∈ Irr(W ) in IndP (ψ), for any ψ ∈ Irr(P ), can be explicitly described by Littlewood–Richardson coeﬃcients.

There is an analogue of this problem which arises in the context of finite reductive groups. Namely, assume W is equipped with an automorphism φ ∈ Aut(W ) stabilizing P . One may consider functions on the cosets P.φ ⊆ P ohφi and W.φ ⊆ W ohφi. Generalizing the notions from finite groups one may consider an irreducible character ψ ∈ Irr(P.φ) on the coset and the corresponding induced character W.φ IndP.φ (ψ). Problem 12. Is there a combinatorial expression, akin to the Littlewood–Richardson coefficient, which W.φ expresses the multiplicity of χ ∈ Irr(W.φ) in IndP.φ (ψ)?

This problem is particularly relevant for determining character values at unipotent elements, as such multiplicities occur in the algorithm mentioned in Theorem 1. An interesting case to consider would be the setting of Theorem 11. For a ﬁxed triple of integers (n, a, b), as in Theorem 11, the multiplicities can be computed using CHEVIE, assuming n is not too large. One could use such software to experimentally test conjectures on a potential combinatorial formula. This could be developed as a research project for undergraduate students.

Lusztig’s Conjecture Recall that if GF is a ﬁnite reductive group then for each χ ∈ Irr(GF ) and X ∈ C(GF ) there exists a F F scalar c(χ, X ) ∈ C which are the entries in the change of basis matrix relating Irr(G ) and C(G ). Determining explicitly these scalars forms a fundamental component of my strategy to determine the multiplicities of irreducible characters in GGGCs. When either determining multiplicities in GGGCs or character values at unipotent elements it is suﬃcient to consider only those scalars c(χ, X ) for which X (u) 6= 0 for some unipotent element u ∈ GF . I am also particularly interested in the case where Z(G) is connected, where one expects a

page 8 Jay Taylor Research Statement complete solution can be achieved. For many applications it is sufficient to consider this theoretically simpler case using the theory of regular embeddings. An approach to determining these scalars, in the unipotently supported case, was developed by Lusztig [Lus86]. There he was able to compute these scalars under restrictions on both G and q. The approach taken by Lusztig is via induction through Levi subgroups. If L 6 G is an F -stable G F F Levi subgroup of G then Deligne–Lusztig have defined a linear map RL : Class(L ) → Class(G ) which takes irreducible characters to virtual characters, which are Z-linear combinations of irreducible characters. G An important part of Lusztig’s inductive argument is being able to compute the map RL . As part of his classification of the irreducible characters of GF [Lus84] Lusztig has introduced a third basis A(GF ) of Class(GF ) given by the almost characters of GF . This basis is defined as an explicit linear combination of the irreducible characters, i.e., it is defined by a change of basis matrix. This change of basis matrix is block diagonal with small blocks. Lusztig’s conjecture then states that for each F F function X ∈ C(G ) there exists a scalar ζ ∈ C× such that ζX ∈ A(G ). Hence, the three bases Irr(GF ), A(GF ), and C(GF ), should be related as follows     A1 0 ζ1 0     Irr(GF )   A(GF )   C(GF )     Irreducible 0 Ak Almost 0 ζt Character Characters Characters Sheaves

To determine the scalars in Lusztig’s conjecture it seems one needs to answer the following problem.

Problem 13. Assume Z(G) is connected. For any F -stable Levi subgroup L 6 G and almost character F G F R ∈ A(L ) determine, explicitly, the decomposition of RL (R) in the basis A(G ) of almost characters.

A potential way to solve this problem relies on the Mackey formula. This is an analogue of the usual G F F Mackey formula for induction and restriction but stated for the map RL : Class(L ) → Class(G ) ∗ G F F and its adjoint RL : Class(G ) → Class(L ). Within the last decade Bonnafé–Michel showed that the Mackey formula holds in all but a few extreme cases [BM11], which occur when q = 2. When Z(G) is connected I was able to push this further to show that the Mackey formula holds unless GF contains a quasi-simple component of type E8(2) [Tay18b]. Assuming the Mackey formula holds I aim to provide a positive answer to Problem 13.

References [BLST17] R. Bezrukavnikov, M. W. Liebeck, A. Shalev, and P. H. Tiep, Character bounds for ﬁnite groups of Lie type, Acta Math. (2017), to appear, arXiv:1707.03896 [math.RT]. [BM11] C. Bonnafé and J. Michel, Computational proof of the Mackey formula for q > 2, J. Algebra 327 (2011), 506–526. [BDT] O. Brunat, O. Dudas, and J. Taylor, Unitriangular shape of decomposition matrices of unipotent blocks, in preperation. [CS17] M. Cabanes and B. Späth, Equivariant character correspondences and inductive McKay condition for type A, J. reine angew. Math. 728 (2017), 153–194. [DLM14] F. Digne, G. I. Lehrer, and J. Michel, On character sheaves and characters of reductive groups at unipotent classes, Pure and Applied Math. Quarterly 10 (2014), 459–512.

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[DM16] O. Dudas and G. Malle, Decomposition matrices for exceptional groups at d = 4, J. Pure Appl. Algebra 220 (2016) no. 3, 1096–1121. [DM18] , Modular irreducibility of cuspidal unipotent characters, Invent. Math. 211 (2018) no. 2, 579–589. [Gec94] M. Geck, Basic sets of Brauer characters of finite groups of Lie type. III, Manuscripta Math. 85 (1994) no. 2, 195–216. [IMN07] I. M. Isaacs, G. Malle, and G. Navarro, A reduction theorem for the McKay conjecture, Invent. Math. 170 (2007) no. 1, 33–101. [Kaw85] N. Kawanaka, Generalized Gelfand-Graev representations and Ennola duality, in: Algebraic groups and related topics (Kyoto/Nagoya, 1983), vol. 6, Adv. Stud. Pure Math. Amsterdam: North- Holland, 1985, 175–206. [Kaw86] , Generalized Gelfand-Graev representations of exceptional simple algebraic groups over a finite field. I, Invent. Math. 84 (1986) no. 3, 575–616. [LOST10] M. W. Liebeck, E. A. O’Brien, A. Shalev, and P. H. Tiep, The Ore conjecture, J. Eur. Math. Soc. (JEMS) 12 (2010) no. 4, 939–1008. [Lus84] G. Lusztig, Characters of reductive groups over a finite field, vol. 107, Annals of Mathematics Studies, Princeton, NJ: Princeton University Press, 1984. [Lus85] , Character sheaves. I, Adv. in Math. 56 (1985) no. 3, 193–237; Character sheaves. II, Adv. in Math. 57 (1985) no. 3, 226–265; Character sheaves. III, Adv. in Math. 57 (1985) no. 3, 266–315; Character sheaves. IV, Adv. in Math. 59 (1986) no. 1, 1–63; Character sheaves. V, Adv. in Math. 61 (1986) no. 2, 103–155. [Lus86] , On the character values of finite Chevalley groups at unipotent elements, J. Algebra 104 (1986) no. 1, 146–194. [Lus92] , A unipotent support for irreducible representations, Adv. Math. 94 (1992) no. 2, 139– 179. [Mic15] J. Michel, The development version of the CHEVIE package of GAP3, J. Algebra 435 (2015), 308– 336. [Nav04] G. Navarro, The McKay conjecture and Galois automorphisms, Ann. of Math. (2) 160 (2004) no. 3, 1129–1140. [SFT18a] A. A. Schaeffer Fry and J. Taylor, On self-normalising Sylow 2-subgroups in type A, J. Lie Theory 28 (2018) no. 1, 139–168. [SFT18b] , Principal 2-blocks and Sylow 2-subgroups, Bull. Lond. Math. Soc. 50 (2018), 733–744. [Sho87] T. Shoji, Green functions of reductive groups over a finite field, in: The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), vol. 47, Proc. Sympos. Pure Math. Prov- idence, RI: Amer. Math. Soc., 1987, 289–301. [Sho07] , Generalized Green functions and unipotent classes for finite reductive groups. II, Nagoya Math. J. 188 (2007), 133–170. [Tay13] J. Taylor, On unipotent supports of reductive groups with a disconnected centre, J. Algebra 391 (2013), 41–61. [Tay14] , Evaluating characteristic functions of character sheaves at unipotent elements, Represent. Theory 18 (2014), 310–340. [Tay15] , Induced characters of type D Weyl groups and the Littlewood–Richardson rule, J. Pure Appl. Algebra 219 (2015) no. 8, 3445–3452. [Tay16] , Generalized Gelfand–Graev representations in small characteristics, Nagoya Math. J. 224 (2016) no. 1, 93–167. [Tay17] , The structure of root data and smooth regular embeddings of reductive groups, Proc. Edinburgh Math. Soc. (2017), to appear, arXiv:1710.05516 [math.RT].

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[Tay18a] , Action of automorphisms on irreducible characters of symplectic groups, J. Algebra 505 (2018), 211–246. [Tay18b] , On the Mackey formula for connected centre groups, J. Group Theory 21 (2018) no. 3, 439–448. [TT18] J. Taylor and P. H. Tiep, Lusztig Induction, Unipotent Supports, and Character Bounds, preprint (2018), arXiv:1809.00173 [math.RT].

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