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Finite Groups of Lie Type: Conjugacy Classes and Complex Characters
... Let G := G(F q ), G := G F = G(F q ), where F is the Frobenius. By abuse of notation, we sometimes identify the group scheme G Fq with its F q -points G. Let G * be the dual group (over F q ) of G, and F * the dual Frobenius (see [Car85,Section 4 ...
... The definition of regular supercuspidal blocks and regular supercuspidal representations of a finite group of Lie type Γ involves modular Deligne-Lusztig theory and block theory. We refer to [DL76], [Car85], and [DM20] for Deligne-Lusztig theory, [BM89] and [Bro90] for modular Deligne-Lusztig theory, and [Bon11, Appendix B] for generalities on blocks. ...
... Recall for s strongly regular semisimple, the (rational) Lusztig series E(G, (s)) consists of only one element, namely, ±R G T (ŝ), whereŝ = θ is such that (T, θ) corresponds to (T * , s) via the bijection in Proposition 3.2.2. Here and after, the sign ± is taken such that ±R G T (ŝ) is an honest representation (see [Car85,Section 7.5]). From now on, we assume moreover that s ∈ G * F * has order prime to ℓ. ...
Let F be a non-archimedean local field with residue characteristic p. Let l be a prime number different from p. Let G be a connected reductive group which is split, semi-simple, and simply connected. On the one hand, we describe the category of quasi-coherent sheaves on the connected component of the stack of L-parameters over Z_l-bar containing a tame, regular semisimple, elliptic L-parameter over F_l-bar. On the other hand, we describe the block of Rep_{Z_l-bar}G(F) containing a depth-zero regular supercuspidal irreducible representation \pi over F_l-bar. For G=GL_n, we compute both sides explicitly and verify the categorical local Langlands conjecture for depth-zero supercuspidal blocks.
... The adjoint G-orbit O(e) of e is uniquely determined by its weighted Dynkin diagram ∆ = ∆(e) which depicts the weights of τ (k × ) on a carefully selected set of simple root vectors of g. These diagrams are the same as in the characteristic zero case and they can be found in [Car93, along with the Dynkin labels of the corresponding nilpotent G-orbits. ...
... On the other hand, dim G e⊗1 = dim g e⊗1 ≥ dim g h⊗1 by Lemma 2.3. Looking through the tables in [Car93, it is now straightforward to see that if e ⊗ 1 ∈ O(A p−1 A r ) for some r ≥ 0, then G is of type E 7 , p = 5 and r = 0. If e ⊗ 1 is not of that type then the sl 2 -triple {e ⊗ 1, h ⊗ 1, f ⊗ 1} must be standard by Remark 2.6. ...
... If (6) holds for ∆ ′ then g h⊗1 ∼ = g(τ, 0) must have type A 4 A 2 . Since dim G e⊗1 ≥ 33, examining the Dynkin diagrams in [Car93,p. 403] one observes that e ⊗ 1 ∈ O(A 3 A 2 A 1 ). If (6) does not hold for ∆ ′ then [Car93, p. 403] reveals that O must have one of the following labels: ...
Let G be an exceptional simple algebraic group over an algebraically closed field k and suppose that the characteristic p of k is a good prime for G. In this paper we classify the maximal Lie subalgebras of the Lie algebra . Specifically, we show that one of the following holds: for some maximal connected subgroup M of G, or is a maximal Witt subalgebra of , or is a maximal \it{\mbox{exotic semidirect product}}. The conjugacy classes of maximal connected subgroups of G are known thanks to the work of Seitz, Testerman and Liebeck--Seitz. All maximal Witt subalgebras of are G-conjugate and they occur when G is not of type and coincides with the Coxeter number of G. We show that there are two conjugacy classes of maximal exotic semidirect products in , one in characteristic 5 and one in characteristic 7, and both occur when G is a group of type .
... No copy of the sign representation occurs in S(h) in degree less that #∆ + c = deg(P K (λ)). The sign representation of W K occurs with multiplicity exactly one in this degree, therefore C[W ] · P K (λ) is irreducible (e.g., [3,Prop. 11.2.3]). ...
... Let σ K denote the W -representation generated by the Weyl dimension polynomial P K (λ) for K. To check whether σ K is a Springer representation for the classical groups, we proceed as follows (see [3,Chapters 11 and 13]): ...
... For example, when G R = SU (p, q), with q ≥ p ≥ 1, the Weyl group W is the symmetric group S p+q , and W K can be identified with the subgroup S p × S q . The representation σ K is parametrized, as a Macdonald representation, by the partition [2 p , 1 q−p ] (see [11] or Proposition 11.4.1 in [3]). This partition corresponds to a 2pq-dimensional nilpotent orbit, so σ K is Springer. ...
Let be a simple real linear Lie group with maximal compact subgroup and assume that . For any representation X of Gelfand-Kirillov dimension , we consider the polynomial on the dual of a compact Cartan subalgebra given by the dimension of the Dirac index of members of the coherent family containing X. Under a technical condition involving the Springer correspondence, we establish an explicit relationship between this polynomial and the multiplicities of the irreducible components occurring in the associated cycle of X. This relationship was conjectured in \cite{MehdiPandzicVogan15}.
... We have a finite expansion (see Equation 3.9.6 below): ...
... where the coefficients s λ,µ ′ are given as p-adic integrals (see Equation 3.9.6 below). By van Leeuwen's formula linking m µ ′ to τ µ with coefficient matrix n µ ′ ,µ , we obtain the coefficients g λ,µ as a matrix product: ...
... Let vol(T ) be the motivic volume of T , viewed as usual as an element of a Grothendieck ring. Upon specialization to a finite field F q , the cardinality of a torus T is given by a determinant [3,Prop.3.3.7]: ...
This article gives a proof of the Langlands-Shelstad fundamental lemma for the spherical Hecke algebra for every unramified p-adic reductive group G in large positive characteristic. The proof is based on the transfer principle for constructible motivic integration. To carry this out, we introduce a general family of partition functions attached to the complex L-group of the unramified p-adic group G. Our partition functions specialize to Kostant's q-partition function for complex connected groups and also specialize to the Langlands L-function of a spherical representation. These partition functions are used to extend numerous results that were previously known only when the L-group is connected (that is, when the p-adic group is split). We give explicit formulas for branching rules, the inverse of the weight multiplicity matrix, the Kato-Lusztig formula for the inverse Satake transform, the Plancherel measure, and Macdonald's formula for the spherical Hecke algebra on a non-connected complex group (that is, non-split unramified p-adic group).
... The result follows. Now let G * be the group dual to G, with dual Frobenius map F * , and write G * = G * F * (see [4,Secs. 4.2 and 4.3]). ...
... We now assume that G has connected center. If s ∈ G * is any semisimple element of G * , then the centralizer C G * (s) is a connected reductive group since we are assuming the center Z(G) is connected [4,Theorem 4.5.9]. So we may consider unipotent characters of the group C G * (s) F * . ...
... We refer to [2, Chapter 15] for a thorough treatment of this bijection, which has many useful properties. The main information we need from this parametrization of characters for G is the description of character degrees, which follows from the main result of Lusztig on the Jordan decomposition of characters [18,Theorem 4.23] (for relevant discussion see [4,Section 12.9] and [6,Remark 13.24]). Suppose that χ ∈ Irr(G) corresponds to the G * -class of pairs (s, ψ), and write χ = χ (s,ψ) . ...
We prove that when q is a power of 2, every complex irreducible representation of may be defined over the real numbers, that is, all Frobenius-Schur indicators are 1. We also obtain a generating function for the sum of the degrees of the unipotent characters of , or of , for any prime power q.
... In particular, technical details of the characters of unitary groups form a substantial part of this work (see Sec. 4). Before proceeding, we briefly recall some results on ordinary representations of finite groups of Lie type (see Carter [9] and Digne and Michel [14]). Let T be an an F -stable maximal torus in G and θ ∈Irr(T F ). ...
... These characters arise from cuspidal unipotent characters of the Levi subgroup D 4 (q) of G . Full details are available in ( [9], Sec. 13.9). ...
... We now turn to the twisted case and complete the proof of Theorem 5.1 when G = 2 E 6 (q). Recall that G F = G = 2 E 6 (q) ad and G = 2 E 6 (q) with q > 2. In the following, φ 8,3 and φ 8,9 denote unipotent characters of G defined in ( [9], Sec. 13.9). ...
For a finite group generated by involutions, the involution width is defined to be the minimal such that any group element can be written as a product of at most k involutions. We show that the involution width of every non-abelian finite simple group is at most 4. This result is sharp, as there are families with involution width precisely 4.
... As in the case of G + , we can identify G − with its dual group (in the sense of the Deligne-Lusztig theory [2], [3]). It is easy to see that any semisimple element s − ∈ G − has centralizer of the form In what follows, we use the convention ...
... Note, see [2,Chapter 13], that if ψ + (λ i ) is the unipotent character of GL k i (q d i ) labeled by the same λ i , then deg ψ + (λ i ) is a product of some powers of q and some cyclotomic polynomials in q; furthermore, if we make the formal change q to −q, then we obtain deg ψ − (λ i ) (up to sign): ...
... . . , r} and for a unique ν ∈ P (2). In this case we denote m j by m ν g and λ j by λ ν g . ...
Let G be a finite symmetric, general linear, or general unitary group defined over a field of characteristic coprime to 3. We construct a canonical correspondence between irreducible characters of degree coprime to 3 of G and those of , where P is a Sylow 3-subgroup of G. Since our bijections commute with the action of the absolute Galois group over the rationals, we conclude that fields of values of character correspondents are the same.
... This case requires additional work. Our notation for finite simple groups (and related ones) follows [C85,Atl]. ...
... By the work of Brunat [B09,Proposition 2], the number of F 0 -invariant p 1 -degree characters of G is equal to the number of semisimple characters of G F 0 " SU n p2q. This number, in turn, is equal to 2 n´1 , by [C85,Corolarry 8.3.6]. Now one can choose the desired χ to be any F 0 -invariant p 1 -degree character of G. ...
We study the sum of the squares of the irreducible character degrees not divisible by some prime p, and its relationship with the the corresponding quantity in a p-Sylow normalizer. This leads to study a recent conjecture by E. Giannelli, which we prove for p=2 and in some other cases.
... The direct calculations of all power X i shows that the rank sequence is always (6,5,4,3,2,1). Letting a, f range through non-zero numbers and b, c, d, e arbitrary one obtains (q − 1) 2 · q 4 matrices. ...
... To obtain a bijection one enriches nilpotent orbits by representations of a suitable group (the original correspondence is obtained by choosing the trivial representations). For the explicit description of the Springer correspondence for g 2 see [8] (Section 7.16) and also [2] (p. 427), [9]. ...
As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of q elements and fixed Jordan type. One obtains polynomials with respect to q with many interesting properties and close relation to type A representation theory. In the present work we develop the corresponding theory for the exceptional Lie algebra . In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.
... In fact, it follows from the Lang-Steinberg theorem (see, for example, [2]) that there exists y ∈ GL(n, K 2 ) such that GL(n, K) F = yGL(n, K) F ′ y −1 . ...
... where g ∈ c µ has Jordan decomposition g = su (thus, by Lemma 3.1 C Un (s) ∼ = L µ ), and Q Lµ Tγ t (u) is a Green function for the unitary group (see, for example, [2]). ...
In his classic book on symmetric functions, Macdonald describes a remarkable result by Green relating the character theory of the finite general linear group to transition matrices between bases of symmetric functions. This connection allows us to analyze the representation theory of the general linear group via symmetric group combinatorics. Using the work of Ennola, Kawanaka, Lusztig and Srinivasan, this paper describes the analogous setting for the finite unitary group. In particular, we explain the connection between Deligne-Lusztig theory and Ennola's efforts to generalize Green's work, and deduce various representation theoretic results from these results. Applications include finding certain sums of character degrees, and a model of Deligne-Lusztig type for the finite unitary group, which parallels results of Klyachko and Inglis and Saxl for the finite general linear group.
... As general references for algebraic groups defined over finite fields we refer the reader to the books by Carter [3] and Digne-Michel [4]. ...
... Using the variation of our program discussed below, one can also show that the number of U(q)-conjugacy classes in the derived subgroup U (1) (q) of U(q) are different for G of types B r and C r , for r = 3, 4, 5. It would be interesting to have a reason for the coincidences in the numbers k(U(q)); we expect it should be explained by the duality of the root systems of type B r and C r , see for example [3,Ch. 4] for similar phenomena. ...
In earlier work, the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a Sylow p-subgroup U(q) of a finite Chevalley group G(q), valid when q is a power of a good prime for G(q). In this paper we develop this algorithm and discuss an implementation in the computer algebra language {\sf GAP}. Using the resulting computer program we are able to calculate the parametrization of the conjugacy classes in U(q), when G(q) is of rank at most 6. In these cases, we observe that the number of conjugacy classes of U(q) is given by a polynomial in q with integer coefficients.
... For root systems of type A and of exceptional type, the tables of component groups in [Car2,§13.1] show that A G (u) is isomorphic to S n with n ≤ 5. Moreover S 4 and S 5 only occur when u is distinguished. For A G (u) ∼ = S 2 and for A G (u) ∼ = S 3 one checks directly that R Z (A G (u)) is torsion-free, by listing all subgroups of A G (u) and all irreducible representations thereof. ...
... is torsion free for all unipotent u ∈ SO 2n+1 (C), which settles the case B n . The root systems of types C n and D n can be handled in a completely analogous way, using the explicit descriptions in [Car2,§13.1]. ...
Let H(R,q) be an affine Hecke algebra with a positive parameter function q. We are interested in the topological K-theory of H(R,q), that is, the K-theory of its C*-completion C*_r (R,q). We will prove that does not depend on the parameter q. For this we use representation theoretic methods, in particular elliptic representations of Weyl groups and Hecke algebras. Thus, for the computation of these K-groups it suffices to work out the case q=1. These algebras are considerably simpler than for q not 1, just crossed products of commutative algebras with finite Weyl groups. We explicitly determine for all classical root data R, and for some others as well. This will be useful to analyse the K-theory of the reduced C*-algebra of any classical p-adic group. For the computations in the case q=1 we study the more general situation of a finite group \Gamma acting on a smooth manifold M. We develop a method to calculate the K-theory of the crossed product . In contrast to the equivariant Chern character of Baum and Connes, our method can also detect torsion elements in these K-groups.
... The necessary background about maximal tori in algebraic groups can all be found in Chapter 3 of Carter [5]. Let us review some of these facts. ...
... Proof: Take q large enough that all maximal tori of G F are non-degenerate, so that the construction of Φ ′ works. From page 29 of Carter [5], a regular semi-simple element α of G lies in a unique maximal torus, which implies by non-degeneracy that α lies in a unique T F . This also implies that ω(α) = Φ ′ (T F ). Therefore: ...
This paper defines and develops cycle indices for the finite classical groups. These tools are then applied to study properties of a random matrix chosen uniformly from one of these groups. Properties studied by this technique will include semisimplicity, regularity, regular semisimplicity, the characteristic polynomial, number of Jordan blocks, and average order of a matrix.
... Here we recall the parametrizations of irreducible representations of Weyl groups for classical types and how they are related to nilpotent orbits in g under the Springer correspondence. One may refer to [Sho79], [Lus79], [Car93] for more information. ...
... (e.g. [Car93,Chapter 13.3]) Therefore, for µ ⊢ n such that µ 1 < λ 1 , we have ch TSp (λ) − Υ λ , ch χ µ = 0. Now we define an ad-hoc notation Res W "Sn" : R(W ) → R(S n ) by (Res W "Sn" f )(w ρ ) = f (w (∅,ρ) ). Then combined with (12.3), we only need to show the following statement. ...
We give explicit formulas on total Springer representations for classical types. We also describe the characters of restrictions of such representations to a maximal parabolic subgroup isomorphic to a symmetric group. As a result, we give closed formulas for the Euler characteristic of Springer fibers.
... There is a criterion for the specialness by inspecting the partitions and their transposes directly, and readers are referred to [40,Proposition 6.3.7]. For the exceptional cases, since there are finitely many of the orbits, it is possible to have exhaustive lists, which can be found for example in [40,43,67]. ...
A bstract
We study a specific type of atomic Higgsings of the 6d N = (1, 0) theories, which we call the induced flows. For the conformal matter theory associated with a pair of nilpotent orbits, the induced flows are given by the inductions of the orbits. We also consider the induced flows for the orbi-instanton theories (as well as some little string theories) that are associated with the homomorphisms from the discrete subgroups of SU(2) to E 8 . This gives a physical definition of the inductions among these discrete homomorphisms, analogous to the inductions of the nilpotent orbits. We analyze the Higgs branch dimensions, the monotonicity of the Weyl anomalies (or the 2-group structure constants for LSTs) and the brane pictures under the induced flows.
... The explicit expressions for the unit quaternions q 1 , q 2 , . . . q 8 that define the representatives of the conjugacy classes of W (F 4 ) are [85]: The characters of W (F 4 ) were computed in [72] (see also [74,86]). We present the characters of W (F 4 ) for the proper rotations in the Table 14 and for the improper rotations in the Table 15. ...
The Yang-Lee universality class arises when imaginary magnetic field is tuned to its critical value in the paramagnetic phase of the Ising model. In d=2, this non-unitary Conformal Field Theory (CFT) is exactly solvable via the M(2,5) minimal model. As found long ago by von Gehlen using Exact Diagonalizations, the corresponding real-time, quantum critical behavior arises in the periodic Ising spin chain when the imaginary longitudinal magnetic field is tuned to its critical value from below. Even though the Hamiltonian is not Hermitian, the energy levels are real due to the PT symmetry. In this paper, we explore the analogous quantum critical behavior in higher dimensional non-Hermitian Hamiltonians on regularized spheres . For d=3, we use the recently invented, powerful fuzzy sphere method, as well as discretization by the platonic solids cube, icosahedron and dodecaherdron. The low-lying energy levels and structure constants we find are in agreement with expectations from the conformal symmetry. The energy levels are in good quantitative agreement with the high-temperature expansions and with Pad\'e extrapolations of the expansions in Fisher's Euclidean field theory for the Yang-Lee criticality. In the course of this work, we clarify some aspects of matching between operators in this field theory and quasiprimary fields in the M(2,5) minimal model. For d=4, we obtain new results by replacing the with the self-dual polytope called the 24-cell.
... In particular we can check if a character corresponds to a Springer representation. Note that the notation for characters of W is slightly different from [7,Chapter 13.3]. The b-invariant of a character can be deduced from [3]. ...
We extend a result of Yun on minimal reduction types to the parahoric case. This implies a uniqueness property for 2-special representations appearing in the cohomology of certain affine Springer fibers. Using this, we settle a conjecture of Lusztig on strata in a reductive group.
... In this note we are interested in Π(GL (F )), namely the "automorphic side" of the Macdonald correspondence (9) for GL (F ), and the main references that we follow closely are Carter (1993);Green (1955) and Macdonald (1980); Ye and Zelingher (2021). ...
Let be a finite group. The trace formula for , which is the trivial case of the Arthur trace formula, is well known with many applications. In this note, we further consider a subgroup Γ of and a representation : Γ → GL() of Γ on a finite dimensional C-vector space , and compute the trace Tr(Ind Γ () ()) of the operator Ind Γ () : Ind Γ () → Ind Γ () for any function : → C in two different ways. The expressions for Tr(Ind Γ () ()) denoted by (,) and (,) are the spectral side and the geometric side of the trace formula for Tr(Ind Γ () ()), respectively. The identity (,) = Tr(Ind Γ () ()) = (,) is a generalization of the trace formula for the finite group. This theory is then applied to the "automorphic side" of the Macdonald correspondence for GL (F); namely, to the "automorphic side" of the local 0-dimensional Langlands correspondence for GL(), where new identities are obtained for the-factors of representations of GL (F). Mathematics Subject Classification (2020): 20C15, 11F72
... There is a criterion for the specialness by inspecting the partitions and their transposes directly, and readers are referred to [40,Proposition 6.3.7]. For the exceptional cases, since there are finitely many of the orbits, it is possible to have exhaustive lists, which can be found for example in [40,43,67]. ...
We study a specific type of atomic Higgsings of the 6d theories, which we call the induced flows. For the conformal matter theory associated with a pair of nilpotent orbits, the induced flows are given by the inductions of the orbits. We also consider the induced flows for the orbi-instanton theories (as well as some little string theories) that are associated with the homomorphisms from the discrete subgroups of to . This gives a physical definition of the inductions among these discrete homomorphisms, analogous to the inductions of the nilpotent orbits. We analyze the Higgs branch dimensions, the monotonicity of the Weyl anomalies (or the 2-group structure constants for LSTs) and the brane pictures under the induced flows.
... Here we summarize some properties of these orbits and corresponding Springer fibers. These can be read from subsequent sections of this paper, [DG84], [Car93], [CM93], etc. ...
We describe Springer fibers corresponding to the minimal and minimal special nilpotent orbits of simple Lie algebras. As a result, we give an answer to the conjecture of Humphreys regarding some graphs attached to Springer fibers.
... Suppose τ is an irreducible representation of W on a (real) vector space V of polynomials in t ∈ R N of dimension n τ . (There is a general result for these groups that real representations suffice, see [1,Chapter 11].) Let P V be the space of polynomial functions R N → V , that is, the generic f ∈ P V can be expressed as f (x, t) where f is a polynomial in x, t and f (x, t) ∈ V for each fixed x ∈ R N . ...
The structure of orthogonal polynomials on with the weight function is based on the Dunkl operators of type . This refers to the full symmetry group of the square, generated by reflections in the lines and . The weight function is integrable if . Dunkl operators can be defined for polynomials taking values in a module of the associated reflection group, that is, a vector space on which the group has an irreducible representation. The unique 2-dimensional representation of the group is used here. The specific operators for this group and an analysis of the inner products on the harmonic vector-valued polynomials are presented in this paper. An orthogonal basis for the harmonic polynomials is constructed, and is used to define an exponential-type kernel. In contrast to the ordinary scalar case the inner product structure is positive only when satisfy . For vector polynomials , the inner product has the form where the matrix function K(x) has to satisfy various transformation and boundary conditions. The matrix K is expressed in terms of hypergeometric functions.
... We name a special piece by giving the Bala-Carter label of the special class it contains. The column labelled "Smooth" lists all special pieces that contain only a single class (this is easily deduced from, say, the partial order diagram of unipotent classes in [13,Chapter 13]). Among the remaining special pieces, those with normal closure in any good characteristic (following Thomsen [35]) are listed in the next column, and those known to have normal closure only in characteristic 0 (following Broer [10] and Sommers [32]) appear in the column after that. ...
Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U be an open subset whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent middle extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent middle extension functor. Under suitable hypotheses, we introduce a construction (called "S2-extension") in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical "S2-ification" of appropriate X. The construction also has applications to the "Macaulayfication" problem, and it is particularly well-behaved when X is Gorenstein. Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown in the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S2-extension to give a uniform construction of the desired variety.
... where δ ·,· is the Kronecker symbol. Also, by [C,Theorem 7.5.1] and [S2, (2.6)] we have ...
We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig's label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large.
... The nilpotent G-orbits in g have been classified in terms of so-called weighted Dynkin diagrams (see [5], [6]). From such a diagram it is straightforward to find a h ∈ C lying in an sl 2 -triple (h, e, f ) (cf. [14]). ...
We describe two algorithms for finding representatives of the nilpotent orbits of a theta-group. The algorithms have been implemented in the computer algebra system GAP (inside the package SLA). We comment on their performance. We apply the algorithms to study the nilpotent orbits of theta-groups, where theta is an N-regular automorphism of a simple Lie algebra of exceptional type.
... To determine a d(C min ) note that in all types but D m and 2 D m , the dual of the special class corresponding to λ is the special class corresponding to the conjugate λ * of λ (see [4, §12.7 and §13.4]). Then the claimed expression for a d(C min ) follows from the centraliser orders given in [4 ...
We use the progenerator constructed in our previous paper to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees.
... We first consider the case where g is not isomorphic to a Lie algebra of type C (in particular, we exclude the case where g is of type B 2 ). Applying [Pan1,Theorem 4.2] or analysing the Dynkin labels of nilpotent G-orbits as presented in [Car, one observes that in this case the cocharacter 2θ ∨ : C × → T is optimal in the sense of the Kempf-Rousseau theory for a nonempty Zariski open subset of g(1). It follows that there exists an sl 2 -triple {ẽ, 2h,f } ⊂ g such that f ∈ g(−1),ẽ ∈ g(1) and the adjoint G 0 orbits ofẽ andf are Zariski open in g(1) and g(−1), respectively. ...
We use affine W-algebras to quantize Mishchenko-Fomenko subalgebras for centralizers of nilpotent elements in simple Lie algebras under certain assumptions that are satisfied for all cases in type A and all minimal nilpotent cases outside type .
... The i-th fundamental weight is denoted by ̟ i . 5 Type of g Degrees of basic invariants A n , n 1 1, 2, . . . , n B n , n 3 1, 3, 4, . . . ...
Let be the centraliser of a nilpotent element e in a finite dimensional simple Lie algebra g of rank l over an algebraically closed field of characteristic 0. We investigate the algebra of symmetric invariants of and prove that if g is of type A or C, then is always a graded polynomial algebra in l variables. We show that this continues to hold for some nilpotent elements in the Lie algebras of other types. In type A we prove that is freely generated by a regular sequence in and describe the tangent cone at e to the nilpotent variety of g.
... To deal with the finitely many families of finite simple groups of Lie type of rank less than r(2/3), we apply Theorem 4.1. The only family of finite simple groups of Lie type whose corresponding Coxeter number is less than 3 is PSL 2 (F q ) (see [2].) Thus, Theorem 4.1 implies that if G is a finite simple group of Lie type of rank less than r(2/3), then ζ G (2/3) − 1 ≪ 1 as well, since such groups G can have one of only finitely many Lie types. ...
Answering a question of Gowers, Tao proved that any contains three-term progressions . Using a modification of Tao's argument, we prove such a mixing result for three-term progressions in all nonabelian finite simple groups except for with an error term that depends on the degree of quasirandomness of the group. This argument also gives an alternative proof of Tao's result when , but with the error term .
... Example 2.7 Suppose that λ = ((4, 3, 1), (3,2)). We shall define the λbitableaux t λ ,t λ , t λ andt λ so that Definition 2.8 Suppose that λ is an a-bicomposition of r. (2) ) be the standard λ-bitableau in which the numbers 1, 2, . . . ...
In this paper we use the Hecke algebra of type B to define a new algebra \Sch which is an analogue of the q-Schur algebra. We construct Weyl modules for \Sch and obtain, as factor modules, a family of irreducible \Sch-modules over any field.
... This section recalls few facts on the Iwahori-Hecke algebras associated with the finite Coxeter groups. The general references on the Hecke algebras are Curtis and Reiner [4], §67-68, Carter [3], §10.8-10.11. We follow the presentation in the paper by Diaconis and Ram [5], which contains the necessary representation theoretic background (sections 3 and 7). ...
In the present paper we construct and solve a differential model for the q-analog of the Plancherel growth process. The construction is based on a deformation of the Makrov-Krein correspondence between continual diagrams and probability distributions.
... The commutator subgroup [U, U] is an F -stable closed connected normal subgroup of U. We define the subgroup U * := [U, U] F ⊆ U ; then [U, U ] ⊆ U * . Furthermore, we shall fix a group homomorphism σ : U → k × which is a regular character, that is, we have U * ⊆ ker(σ) and the restriction of σ to U s is non-trivial for all s ∈ S. (Such characters always exist; see [1,Section 8.1] and [2,Definition 14.27].) Then the corresponding module Γ σ = kGu σ is called a Gelfand-Graev module for G. Let h ∈ H and σ h : U → k × be defined by σ h (u) := σ(h −1 uh) for u ∈ U . ...
James' submodule theorem is a fundamental result in the representation theory of the symmetric groups and the finite general linear groups. In this note we consider a version of that theorem for a general finite group with a split BN-pair. This gives rise to a distinguished composition factor of the Steinberg module, first described by Hiss via a somewhat different method. It is a major open problem to determine the dimension of this composition factor.
... Proof. By Lemma 3.3 we have (5) IndG G (χ) = IndG G (R G L (ϑ)) = RG L (IndL L (ϑ)). The restriction of every irreducible character ofG to G (and ofL to L) is multiplicity free (see [21,Section 10] and [4,Proposition 15.11]). ...
This work completes the classification of the imprimitive irreducible modules, over algebraically closed fields of characteristic 0, of the finite quasisimple groups.
... In that case, the endomorphism algebra is precisely the Hecke algebra H q (S n ) where q is the order of the finite field, exactly analogous to the case of P KZ = Ind W 1 C for rational Cherednik algebras. In the general case, Howlett and Lehrer [HL,Theorem 4.14] showed that, in characteristic 0, the endomorphism algebra of a parabolically induced cuspidal representation of a finite group of Lie type can be described as a semidirect product of a finite type Hecke algebra by a finite group acting by a diagram automorphism, twisted by a certain 2-cocycle (see, for instance, [Car,Theorem 10.8.5]). We obtain an exactly analogous result for the endomorphism algebras of induced representations Ind W W ′ L of finite-dimensional irreducible representations L of H c (W ′ , h W ′ ). ...
For a complex reflection group W with reflection representation , we define and study a natural filtration by Serre subcategories of the category of representations of the rational Cherednik algebra . This filtration refines the filtration by supports and is analogous to the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. Using the monodromy of the Bezrukavnikov-Etingof parabolic restriction functors, we show that the subquotients of this filtration are equivalent to categories of finite-dimensional representations over generalized Hecke algebras. When W is a finite Coxeter group, we give a method for producing explicit presentations of these generalized Hecke algebras in terms of finite-type Iwahori-Hecke algebras. This yields a method for counting the number of irreducible objects in of given support. We apply these techniques to count the number of irreducible representations in of given support for all exceptional Coxeter groups W and all parameters c, including the unequal parameter case. This completes the classification of the finite-dimensional irreducible representations of for exceptional Coxeter groups W in many new cases.
... For example, in the case of the group G = GL n (F q ) and its Borel subgroup B of upper triangular matrices, the group W is the symmetric group. There are many wonderful references for this material; Brown [Bw], Bourbaki [Bou,, Curtis and Reiner [CR,[67][68] or Carter [Ca,. We develop what we need in this section and give the relation to probability theory. ...
Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques
We solve two open problems in Coxeter–Catalan combinatorics. First, we introduce a family of rational noncrossing objects for any finite Coxeter group, using the combinatorics of distinguished subwords. Second, we give a type‐uniform proof that these noncrossing Catalan objects are counted by the rational Coxeter–Catalan number, using the character theory of the associated Hecke algebra and the properties of Lusztig's exotic Fourier transform. We solve the same problems for rational noncrossing parking objects.
We study representations of that are distinguished with respect to a symmetric subgroup , where is an involution. We prove that those representations satisfy , thus positively answering a version of the Prasad-Lapid conjecture.
We define an involution on the elliptic space of tempered unipotent representations of inner twists of a split simple p -adic group G and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact open subgroups. In particular, we formulate a precise conjecture about the relation with a version of Lusztig's nonabelian Fourier transform on the space of unipotent representations of the (possibly disconnected) reductive quotients of maximal compact subgroups. We give evidence for the conjecture, including proofs for and .
Given a complex representation of a finite group, Brauer and Nesbitt defined in 1941 its reduction mod p, obtaining a representation over the algebraic closure of . In 2021, Lusztig studied the characters obtained by reducing mod p an irreducible unipotent representation of a finite reductive group over . He gave a conjectural formula for this character as a linear combination of terms which had no explicit definition and were only known in some small-rank examples. In this paper we provide an explicit formula for these terms and prove Lusztig's conjecture, giving a formula for the reduction mod p of any unipotent representation of for q a power of p. We also propose a conjecture linking this construction to the full exceptional collection in the derived category of coherent sheaves on a partial flag variety constructed recently by Samokhin and van der Kallen.
Let be a connected reductive algebraic group over an algebraically closed field and be a Borel subgroup of . In this paper, we completely determine the composition factors of the permutation module for any field .
Every four years leading researchers gather to survey the latest developments in all aspects of group theory. Since 1981, the proceedings of these meetings have provided a regular snapshot of the state of the art in group theory and helped to shape the direction of research in the field. This volume contains selected papers from the 2022 meeting held in Newcastle. It includes substantial survey articles from the invited speakers, namely the mini course presenters Michel Brion, Fanny Kassel and Pham Huu Tiep; and the invited one-hour speakers Bettina Eick, Scott Harper and Simon Smith. It features these alongside contributed survey articles, including some new results, to provide an outstanding resource for graduate students and researchers.
We propose upper bounds for the number of modular constituents of the restriction modulo p of a complex irreducible character of a finite group, and for its decomposition numbers, in certain cases.
In this paper, we show that the elliptic cocenter of the Hecke algebra of a connected reductive group over a nonarchimedean local field is contained in the rigid cocenter. As applications, we prove the trace Paley-Wiener theorem and the abstract Selberg principle for mod-l representations.
One of the main problems in representation theory is to understand the exact relationship between Brauer corresponding blocks of finite groups. The case where the local correspondent has a unique simple module seems key. We characterize this situation for the principal p-blocks where p is odd.
The Cartan scheme of a finite group G with a (B,N)-pair is defined to be the coherent configuration associated with the action of G on the right cosets of the Cartan subgroup by the right multiplications. It is proved that if G is a simple group of Lie type, then asymptotically, the coherent configuration is 2-separable, i.e., the array of 2-dimensional intersection numbers determines up to isomorphism. It is also proved that in this case, the base number of equals 2. This enables us to construct a polynomial-time algorithm for recognizing the Cartan schemes when the rank of G and order of the underlying field are sufficiently large. One of the key points in the proof of the main results is a new sufficient condition for an arbitrary homogeneous coherent configuration to be 2-separable.
Let G be a group. Two elements are said to be in the same z-class if their centralizers in G are conjugate within G. Consider a perfect field of characteristic , which has a non-trivial Galois automorphism of order 2. Further, suppose that the fixed field has the property that it has only finitely many field extensions of any finite degree. In this paper, we prove that the number of z-classes in the unitary group over such fields is finite. Further, we count the number of z-classes in the finite unitary group , and prove that this number is same as that of when .
For any grading by an abelian group G on the exceptional simple Lie algebra of type or over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of G-graded simple -modules, as well as necessary and sufficient conditions for an -module to admit a G-grading compatible with the given G-grading on .
In the paper four stratifications in the reduction modulo p of a general Shimura variety are studied: the Newton stratification, the Kottwitz-Rapoport stratification, the Ekedahl-Oort stratification and the Ekedahl-Kottwitz-Oort-Rapoport stratification. We formulate a system of axioms and show that these imply non-emptiness statements and closure relation statements concerning these various stratifications. These axioms are satisfied in the Siegel case.
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