The Hecke category is at the heart of several fundamental questions in modular representation theory. We emphasise the role of the "philosophy of deformations" both as a conceptual and computational tool, and suggest possible connections to Lusztig's "philosophy of generations". On the geometric side one can understand deformations in terms of localisation in equivariant cohomology. Recently Treumann and Leslie-Lonergan have added Smith theory, which provides a useful tool when considering mod p coefficients. In this context, we make contact with some remarkable work of Hazi. Using recent work of Abe on Soergel bimodules, we are able to reprove and generalise some of Hazi's results. Our aim is to convince the reader that the work of Hazi and Leslie-Lonergan can usefully be viewed as some kind of localisation to "good" reflection subgroups. These are notes for my lectures at the 2019 Current Developments in Mathematics at Harvard.
Using Cohen's classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group are themselves symplectic reflection groups. This is the symplectic analogue of Steinberg's Theorem for complex reflection groups. Using computational results required in the proof, we show the non-existence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open. An…
Let $\ell$ be a prime divisor of the order of a finite unitary reflection group. We classify up to conjugacy the parabolic and reflection subgroups that are minimal with respect to inclusion, subject to containing an $\ell$-Sylow subgroup. The classification assists in describing the $\ell$-Sylow subgroups of unitary reflection groups up to group isomorphism. This classification also relates to the modular representation theory of finite groups of Lie type. We observe that unless a parabolic subgroup minimally containing an $\ell$-Sylow subgroup is $G$ itself, the reflection subgroup within th…
We study normal reflection subgroups of complex reflection groups. Our point of view leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalized exponents. Our refinement gives a uniform proof and generalization of a recent theorem of the second author.
In this paper, we determine the modular invariants of finite modular pseudo-reflection subgroups of the finite general linear group $ \text{GL}_n(q) $ acting on the tensor product of the symmetric algebra $ S^{\bullet}(V) $ and the exterior algebra $ \wedge^{\bullet}(V) $ of the natural $ \text{GL}_n(q) $-module $ V $. We are particularly interested in the case where $ G $ is a subgroup of the parabolic subgroups of $ \text{GL}_n(q) $ which is a generalization of Weyl group of Cartan type Lie algebra.
It is known that the Selberg zeta function for the modular group has an expression in terms of the class numbers and the fundamental units of the indefinite binary quadratic forms. In the present paper, we generalize such a expression to any congruence subgroup of the modular groups in SL(2,R) and SL(2,C).
Beijing International Center for Mathematical Research Peking University, Beijing, China（北京大学北京国际数学研究中心）；School of Mathematics and Statistics University of Melbourne, VIC 3010, Australia（墨尔本大学数学与统计学院）；Department of Foundational Mathematics School of Mathematics and Physics Xi’an Jiaotong-Liverpool University, Suzhou, China（西交利物浦大学数学与物理学院基础数学系）
As a sequel of \cite{Ou}, in this shot note, we investigate block basis for coinvariants of finite modular pseudo-reflection groups. We are particularly interested in the case where $ G $ is a subgroup of the parabolic subgroups of $ GL_n(q) $ which generalizes the Weyl groups of restricted Cartan type Lie algebra.
Colmez has given a recipe to associate a smooth modular representation Omega(W) of the Borel subgroup of GL_2(Q_p) to a F_p^bar-representation W of Gal(Qp^bar/Qp) by using Fontaine's theory of (phi,Gamma)-modules. We compute Omega(W) explicitly and we prove that if W is irreducible and dim(W)=2, then Omega(W) is the restriction to the Borel subgroup of GL_2(Q_p) of the supersingular representation associated to W in Breuil's correspondence.
Let $W_0$ be a reflection subgroup of a finite complex reflection group $W$, and let $B_0$ and $B$ be their respective braid groups. In order to construct a Hecke algebra $\widetilde{H}_0$ for the normalizer $N_W(W_0)$, one first considers a natural subquotient $\widetilde{B}_0$ of $B$ which is an extension of $N_W(W_0)/W_0$ by $B_0$. We prove that this extension is split when $W$ is a Coxeter group, and deduce a standard basis for the Hecke algebra $\widetilde{H}_0$. We also give classes of both split and non-split examples in the non-Coxeter case.
The imaginary cone of a Kac-Moody Lie algebra is the convex hull of zero and the positive imaginary roots. This paper studies the imaginary cone for a class of root systems of general Coxeter groups W. It is shown that the imaginary cone of a reflection subgroup of W is contained in that of W, and that for irreducible infinite W of finite rank, the closed imaginary cone is the only non-zero, closed, pointed W-stable cone contained in the pointed cone spanned by the simple roots. For W of finite rank, various natural notions of faces of the imaginary cone are shown to coincide, the face lattice…
Broué, Malle and Michel have shown that the automizer of an abelian Sylow p-subgroup in a finite simple Chevalley group is an irreducible complex reflection group, for p not too small and different from the defining characteristic. The aim is this note is to show that a suitable version of this property holds for general finite groups. As an example, the automizer of an 11-Sylow subgroup in the Monster is the 2-dimensional complex reflection group G_{16}.
This article extends the works of Gonçalves, Guaschi, Ocampo [GGO] and Marin [MAR2] on finite subgroups of the quotients of generalized braid groups by the derived subgroup of their pure braid group. We get explicit criteria for subgroups of the (complex) reflection group to lift to subgroups of this quotient. In the specific case of the classical braid group, this enables us to describe all its finite subgroups : we show that every odd-order finite group can be embedded in it, when the number of strands goes to infinity. We also determine a complete list of the irreducible reflection groups f…
Let E be a finite extension of Fp. Using Fontaine's theory of (phi,Gamma)-modules, Colmez has shown how to attach to any irreducible E-linear representation of Gal(Qpbar/Qp) an infinite dimensional smooth irreducible E-linear representation of B_2(Qp) that has a central character. We prove that every such representation of B_2(Qp) arises in this way. Our proof extends to algebraically closed fields E of characteristic p. In this case, infinite dimensional smooth irreducible E-linear representations of B_2(Qp) having a central character arise in a similar way from irreducible E-linear represent…
We prove the following theorem. Let $G$ be a finite group generated by unitary reflections in a complex Hermitian space $V=\mathbb{C}^\ell$ and let $G'$ be any reflection subgroup of $G$. Let $\mathcal{H}(G)$ be the space of $G$-harmonic polynomials on $V$. There is a degree preserving isomorphism $ξ:\mathcal{H}(G')\otimes\mathcal{H}(G)^{G'}\overset{\sim}{\longrightarrow}\mathcal{H}$ of graded $\mathcal{N}$-modules, where $\mathcal{N}:=N_{\rm{GL}(V)}(G)\cap N_{\rm{GL}(V)}(G')$ and $\mathcal{H}^{G'}$ is the space of $G'$-fixed points of $\mathcal{H}$. This generalises a result of Douglass and D…
In the context of Hecke algebras of complex reflection groups, we prove that the generalized Hecke algebras of normalizers of parabolic subgroups are semidirect products, under suitable conditions on the parameters involved in their definition.
In 1878, Jordan showed that a finite subgroup of GL(n,C) contains an abelian normal subgroup whose index is bounded by a function of n alone. Previously, the author has given precise bounds. Here, we consider analogues for finite linear groups over algebraically closed fields of positive characteristic l. A larger normal subgroup must be taken, to eliminate unipotent subgroups and groups of Lie type and characteristic l, and we show that generically the bound is similar to that in characteristic 0 - being (n+1)!, or (n+2)! when l divides (n+2) - given by the faithful representations of minimal…
The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of any rational, C_2-cofinite vertex operator algebra is a congruence subgroup. In particular, the q-character of each irreducible module is a modular function on the same congruence subgroup. The Galois symmetry of the modular representations is obtained and the order of the anomaly for those modular categories satisfying some integrality conditions is determine…
For each subgroup of GL_2(F_p) or order divisible by p, generated by (pseudo-)reflections, we compute the ideals of stable and generalized invariants. These groups and these ideals are related to the cohomology of compact Lie groups, Kac-Moody groups and p-compact groups.
We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $Γ$ be a finite index subgroup of $\mathrm{SL}(2,\mathbb{Z})$ with an action on a complex simple Lie algebra $\mathfrak g$, which can be extended to $\mathrm{SL}(2,\mathbb{C})$. We show that the Lie algebra of the corresponding $\mathfrak{g}$-valued modular forms is isomorphic to the extension of $\mathfrak{g}$ over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups $Γ(N), \,…
We prove that the kernel of the natural action of the modular group on the center of the Drinfel'd double of a semisimple Hopf algebra is a congruence subgroup. To do this, we introduce a class of generalized Frobenius-Schur indicators and endow it with an action of the modular group that is compatible with the original one.
