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 gen- eralized 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 homo- morphisms 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 character- istic 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 ) 2 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 2 Irr(GF ) and characteristic function X 2 C(GF ) there exists a scalar c(χ, X ) 2 C such that X χ = c(χ, X )X : X 2C(GF ) page 2 Jay Taylor Research Statement Computing the value of an irreducible character χ 2 Irr(GF ) at an element g 2 GF can be viewed as the following two step process: 1) for any function X 2 C(GF ) compute the value X (g), 2) for any X 2 C(GF ) with X (g) 6= 0 compute the scalar c(χ, X ). That the values of X 2 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 2 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 2 C(GF ) and unipotent element u 2 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 2 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.