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Let $G$ be a subgroup of ${\rm PGL}_2({\mathbb F}_q)$, where $q$ is any prime power, and let $Q \in {\mathbb F}_q[x]$ such that ${\mathbb F}_q(x)/{\mathbb F}_q(Q(x))$ is a Galois extension with group $G$. By explicitly computing the Artin map on unramified degree-1 primes in ${\mathbb F}_q(Q)$ for various groups $G$, interesting new results emerge about finite fields, additive polynomials, and conjugacy classes of ${\rm PGL}_2({\mathbb F}_q)$. For example, by taking $G$ to be a unipotent group, one obtains a new characterization for when an additive polynomial splits completely over ${\mathbb F}_q$. When $G = {\rm PGL}_2({\mathbb F}_q)$, one obtains information about conjugacy classes of ${\rm PGL}_2({\mathbb F}_q)$. When $G$ is the group of order 3 generated by $x \mapsto 1 - 1/x$, one obtains a natural tripartite symbol on ${\mathbb F}_q$ with values in ${\mathbb Z}/3{\mathbb Z}$. Some of these results generalize to ${\rm PGL}_2(K)$ for arbitrary fields $K$. Apart from the introduction, this article is written from first principles, with the aim to be accessible to graduate students or advanced undergraduates. An earlier draft of this article was published on the Math arXiv in June 2019 under the title {\it More structure theorems for finite fields}., Comment: Version 4 contains minor corrections and updates to the bibliograpy. Version 3 is a major revision, including a change in the title from "More structure theorems for finite fields" to "Explicit Artin maps into PGL2". The author thanks Xander Faber for insightful comments that led to the change in the title

A Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares with conjugate symmetry and classify them according to several common notions of equivalence. We also do similar enumerations under additional hypotheses, such as assuming the Latin square is reduced, diagonal, idempotent or unipotent. Our data corrected an error in earlier literature and suggested several patterns that we then found proofs for, including (1) The number of isomorphism classes of semisymmetric idempotent Latin squares of order $n$ equals the number of isomorphism classes of semisymmetric unipotent Latin squares of order $n+1$, and (2) Suppose $A$ and $B$ are totally symmetric Latin squares of order $n\not\equiv0\bmod3$. If $A$ and $B$ are paratopic then $A$ and $B$ are isomorphic.

We prove a certain transcendence property of the unipotent Albanese map of a smooth variety, conditional on the Ax-Schanuel conjecture for variations of mixed Hodge structure. We show that this property allows the Chabauty-Kim method to be generalized to higher-dimensional varieties. In particular, we conditionally generalize several of the main Diophantine finiteness results in Chabauty-Kim theory to arbitrary number fields., 14 pages

In this paper, we construct and classify the special unipotent representations of a real reductive group attached to the principal nilpotent orbit. We give formulas for the $\mathbf{K}$-types, associated varieties, and Langlands parameters of all such representations., revised exposition

The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of $Rep_R(G)$, the category of smooth representations of a $p$-adic group $G$ with coefficients in $R$. In particular, they were used to construct level 0 decompositions when $R=\overline{\mathbb{Z}}_{\ell}$, $\ell \neq p$, by Dat for $GL_n$ and the author for a more general group. Wang proved in the case of $GL_n$ that the subcategory associated with a system of idempotents is equivalent to a category of coefficient systems on the Bruhat-Tits building. This result was used by Dat to prove an equivalence between an arbitrary level zero block of $GL_n$ and a unipotent block of another group. In this paper, we generalize Wang's equivalence of category to a connected reductive group on a non-archimedean local field., 17 pages, in English

For a general linear supergroup $G=GL(m|n)$, we consider a natural isomorphism $\phi: G \to U^-\times G_{ev} \times U^+$, where $G_{ev}$ is the even subsupergroup of $G$, and $U^-$, $U^+$ are appropriate odd unipotent subsupergroups of $G$. We compute the action of odd superderivations on the images $\phi^*(x_{ij})$ of the generators of $K[G]$. We describe a specific ordering of the dominant weights $X(T)^+$ of $GL(m|n)$ for which there exists a Donkin-Koppinen filtration of the coordinate algebra $K[G]$. Let $\Gamma$ be a finitely generated ideal $\Gamma$ of $X(T)^+$ and $O_{\Gamma}(K[G])$ be the largest $\Gamma$-subsupermodule of $K[G]$ having simple composition factors of highest weights $\lambda\in \Gamma$. We apply combinatorial techniques, using generalized bideterminants, to determine a basis of $G$-superbimodules appearing in Donkin-Koppinen filtration of $O_{\Gamma}(K[G])$.

We examine the cohomology and representation theory of a family of finite supergroup schemes of the form $(\mathbb G_a^-\times \mathbb G_a^-)\rtimes (\mathbb G_{a(r)}\times (\mathbb Z/p)^s)$. In particular, we show that a certain relation holds in the cohomology ring, and deduce that for finite supergroup schemes having this as a quotient, both cohomology mod nilpotents and projectivity of modules is detected on proper sub-super\-group schemes. This special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes, in a separate work of the authors joint with Iyengar and Krause. We also completely determine the cohomology ring in the smallest cases, namely $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb G_{a(1)}$ and $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb Z/p$. The computation uses the local cohomology spectral sequence for group cohomology, which we describe in the context of finite supergroup schemes., 19 pages

S.-Y. Pan decomposes the uniform projection of the Weil representation of a finite symplectic-odd orthogonal dual pair, in terms of Deligne-Lusztig virtual characters, assuming that the order of the finite field is large enough. In this paper we use Pan's decomposition to study the theta correspondence for this kind of dual pairs, following the approach of Adams-Moy and Aubert-Michel-Rouquier. Our results give the theta correspondence between unipotent representations and certain quadratic unipotent representations., Comment: Revised version, accepted by Pacific J. Math

Let G(F_q) be the group of rational points of a simple algebraic group defined and split over a finite field F_q. In this paper we define a new basis for the Grothendieck group of unipotent representations of G(F_q)., 36 pages. arXiv admin note: text overlap with arXiv:1805.03770

To each automorphism of a spherical building, there is a naturally associated opposition diagram, which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber, then the automorphism is called domestic. In this paper, we give the complete classification of domestic automorphisms of split spherical buildings of types $\mathsf {E}_6$ , $\mathsf {F}_4$ , and $\mathsf {G}_2$ . Moreover, for all split spherical buildings of exceptional type, we classify (i) the domestic homologies, (ii) the opposition diagrams arising from elements of the standard unipotent subgroup of the Chevalley group, and (iii) the automorphisms with opposition diagrams with at most two distinguished orbits encircled. Our results provide unexpected characterizations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.

This article deals with the tamely ramified geometric Langlands correspondence for GL_2 on $\mathbf{P}_{\mathbf{F}_q}^1$, where $q$ is a prime power, with tame ramification at four distinct points $D = \{\infty, 0,1, t\} \subset \mathbf{P}^1(\mathbf{F}_q)$. We describe in an explicit way (1) the action of the Hecke operators on a basis of the cusp forms, which consists of $q$ elements; and (2) the correspondence that assigns to a pure irreducible rank 2 local system $E$ on $\mathbf{P}^1 \setminus D$ with unipotent monodromy its Hecke eigensheaf on the moduli space of rank 2 parabolic vector bundles. We define a canonical embedding $\mathbf{P}^1$ into this module space and show with a new proof that the restriction of the eigensheaf to the degree 1 part of this moduli space is the intermediate extension of $E$., 34 pages

Falbel, Koseleff and Rouillier computed a large number of boundary unipotent CR representations of fundamental groups of non compact three-manifolds. Those representations are not always discrete. By experimentally computing their limit set, one can determine that those with fractal limit sets are discrete. Many of those discrete representations can be related to (3,3,n) complex hyperbolic triangle groups. By exact computations, we verify the existence of those triangle representations, which have boundary unipotent holonomy. We also show that many representations are redundant: for n fixed, all the (3,3,n) representations encountered are conjugate and only one among them is uniformizable.

We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the $p$-adic de Rham period map $j^{dr}_n$ on elliptic and hyperelliptic curves over number fields via their universal unipotent connections $\mathcal{U}$. Several algorithms forming part of the computation of finite level versions $j^{dr}_n$ of the unipotent Albanese maps are presented. The computation of the logarithmic extension of $\mathcal{U}$ in general requires a description in terms of an open covering, and can be regarded as a simple example of computational descent theory. We also demonstrate a constructive version of a lemma of Hadian used in the computation of the Hodge filtration on $\mathcal{U}$ over affine elliptic and odd hyperelliptic curves. We use these algorithms to present some new examples describing the co-ordinates of some of these period maps. This description will be given in terms iterated $p$-adic Coleman integrals. We also consider the computation of the co-ordinates if we replace the rational basepoint with a tangential basepoint, and present some new examples here as well., Comment: 60 pages

Let $G$ be a finite Chevalley group. We are concerned with computing the values of the unipotent characters of $G$ by making use of Lusztig's theory of character sheaves. In this framework, one has to find the transformation between several bases for the class functions on $G$. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. We shall determine these scalars in the specific case where $G$ is the (twisted or non-twisted) group of type $E_6$ over the finite field with $q$ elements, for $q$ a power of the bad prime $p=3$, by exploiting known facts about the representation theory of the Hecke algebra associated with $G$., Comment: 12 pages

This paper has two main parts. Firstly, we give a classification of the $\ell$-blocks of finite special linear and unitary groups $SL_n(\epsilon q)$ in the non-defining characteristic $\ell\ge 3$. Secondly, we describe how the $\ell$-weights of $SL_n(\epsilon q)$ can be obtained from the $\ell$-weights of $GL_n(\epsilon q)$ when $\ell\nmid\mathrm{gcd}(n,q-\epsilon)$, and verify the Alperin weight conjecture for $SL_n(\epsilon q)$ under the condition $\ell\nmid\mathrm{gcd}(n,q-\epsilon)$. As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for any unipotent $\ell$-block of $SL_n(\epsilon q)$ if $\ell\nmid\mathrm{gcd}(n,q-\epsilon)$., Comment: revised version, to appear in Journal of Algebra

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (resp. subalgebra), or a natural generalisation if $G$ is of type $D_n$. From these representatives, it is straightforward to generate representatives for any given class. Along the way we also prove recognition theorems for identifying both the unipotent classes and nilpotent orbits of exceptional algebraic groups., 26 pages

This work concerns the representation theory and cohomology of a finite unipotent supergroup scheme $G$ over a perfect field $k$ of positive characteristic $p\ge 3$. It is proved that an element $x$ in the cohomology of $G$ is nilpotent if and only if for every extension field $K$ of $k$ and every elementary sub-supergroup scheme $E\subseteq G_K$, the restriction of $x_K$ to $E$ is nilpotent. It is also shown that a $kG$-module $M$ is projective if and only if for every extension field $K$ of $k$ and every elementary sub-supergroup scheme $E\subseteq G_K$, the restriction of $M_K$ to $E$ is projective. The statements are motivated by, and are analogues of, similar results for finite groups and finite group schemes, but the structure of elementary supergroups schemes necessary for detection is more complicated than in either of these cases. One application is a detection theorem for the nilpotence of cohomology, and projectivity of modules, over finite dimensional Hopf subalgebras of the Steenrod algebra., 46 pages; Sections 12 on Z-graded group schemes and the Steenrod algebra is revised compared to the previous version

Let $V$ be a finite-dimensional vector space over a field of characteristic two. As the main result of this paper, for every nilpotent element $e \in \mathfrak{sl}(V)$, we describe the Jordan normal form of $e$ on the $\mathfrak{sl}(V)$-modules $\wedge^2(V)$ and $S^2(V)$. In the case where $e$ is a regular nilpotent element, we are able to give a closed formula. We also consider the closely related problem of describing, for every unipotent element $u \in \operatorname{SL}(V)$, the Jordan normal form of $u$ on $\wedge^2(V)$ and $S^2(V)$. A recursive formula for the Jordan block sizes of $u$ on $\wedge^2(V)$ was given by Gow and Laffey (J. Group Theory 9 (2006), 659-672). We show that their proof can be adapted to give a similar formula for the Jordan block sizes of $u$ on $S^2(V)$., to appear in Linear Algebra and its Applications

Our computations show that there is a total of $40$ pairs of degree six coprime polynomials $f,g$ where $f(x)=(x-1)^6$, $g$ is a product of cyclotomic polynomials, $g(0)=1$ and $f,g$ form a primitive pair. The aim of this article is to determine whether the corresponding $40$ symplectic hypergeometric groups with a maximally unipotent monodromy follow the same dichotomy between arithmeticity and thinness that holds for the $14$ symplectic hypergeometric groups corresponding to the pairs of degree four polynomials $f,g$ where $f(x)=(x-1)^4$ and $g$ is as described above. As a result we prove that at least $18$ of these $40$ groups are arithmetic in $\mathrm{Sp}(6)$. In addition, we extend our search to all degree six symplectic hypergeometric groups. We find that there is a total of $458$ pairs of polynomials (up to scalar shifts) corresponding to such groups. For $211$ of them, the absolute values of the leading coefficients of the difference polynomials $f-g$ are at most $2$ and the arithmeticity of the corresponding groups follows from Singh and Venkataramana, while the arithmeticity of one more hypergeometric group follows from Detinko, Flannery and Hulpke. In this article, we show the arithmeticity of $160$ of the remaining $246$ hypergeometric groups., Comment: 25 Pages, 4 Tables. Author's list has been updated. The article has been reorganized. New results have been found and incorporated

We give a new realization of the prefundamental representations $L^\pm_{r,a}$ introduced by Hernandez and Jimbo, when the quantum loop algebra $U_q(\mathfrak{g})$ is of types $A_n^{(1)}$ and $D_n^{(1)}$, and the $r$-th fundamental weight $\varpi_r$ for types $A_n$ and $D_n$ is minuscule. We define an action of the Borel subalgebra $U_q(\mathfrak{b})$ of $U_q(\mathfrak{g})$ on the unipotent quantum coordinate ring associated to the translation by $-\varpi_r$, and show that it is isomorphic to $L^\pm_{r,a}$. We then give a combinatorial realization of $L^+_{r,a}$ in terms of the Lusztig data of the dual PBW vectors., Comment: 41 pages, introduction is revised, Lemma 3.2 is revised, the proofs of Lemma 3.8 and Corollary 4.8 are revised, added Example 3.9, Example 4.4 and Example 4.19, comments added on Remark 3.11, Remark 4.9, Remark 4.21 and Remark 4.30, typos corrected, minor corrections

Let u be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at u with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition which is part of the cell decomposition (Rietsch) of the totally positive flag manifold., Comment: 20 pages. A new section has been added

Using the geometric Satake correspondence, the Mirkovic-Vilonen cycles in the affine Grasssmannian give bases for representations of a semisimple group G . We prove that these bases are "perfect", i.e. compatible with the action of the Chevelley generators of the positive half of the Lie algebra g. We compute this action in terms of intersection multiplicities in the affine Grassmannian. We prove that these bases stitch together to a basis for the algebra C[N] of regular functions on the unipotent subgroup. We compute the multiplication in this MV basis using intersection multiplicities in the Beilinson-Drinfeld Grassmannian, thus proving a conjecture of Anderson. In the third part of the paper, we define a map from C[N] to a convolution algebra of measures on the dual of the Cartan subalgebra of g. We characterize this map using the universal centralizer space of G. We prove that the measure associated to an MV basis element equals the Duistermaat-Heckman measure of the corresponding MV cycle. This leads to a proof of a conjecture of Muthiah. Finally, we use the map to measures to compare the MV basis and Lusztig's dual semicanonical basis. We formulate conjectures relating the algebraic invariants of preprojective algebra modules (which underlie the dual semicanonical basis) and geometric invariants of MV cycles. In the appendix, we use these ideas to prove that the MV basis and the dual semicanonical basis do not coincide in SL_6., 82 pages, with an appendix by Calder Morton-Ferguson and Anne Dranowski

Dual canonical bases are expected to satisfy a certain (double) triangularity property by Leclerc's conjecture. We propose an analogous conjecture for common triangular bases of quantum cluster algebras. We show that a weaker form of the analogous conjecture is true. Our result applies to the dual canonical bases of quantum unipotent subgroups. It also applies to the t-analogs of q-characters of simple modules of quantum affine algebras.

We construct a categorification of the modular data associated with every family of unipotent characters of the spetsial complex reflection group $G(d,1,n)$. The construction of the category follows the decomposition of the Fourier matrix as a Kronecker tensor product of exterior powers of the character table $S$ of the cyclic group of order $d$. The representation of the quantum universal enveloping algebra of the general linear Lie algebra $\mathfrak{gl}_m$, with quantum parameter an even root of unity of order $2d$, provides a categorical interpretation of the matrix $\bigwedge^m S$. We also prove some positivity conjectures of Cuntz at the decategorified level., 27 pages, comments welcome

Let $G$ be a connected semisimple Lie group with finite centre, and let $M= \Gamma \backslash G$ be a compact homogeneous manifold. Under a spectral gap assumption, we show that smooth time-changes of any unipotent flow on $M$ have polynomial decay of correlations. Our result applies also in the case where $M$ is a finite volume, non-compact quotient under some additional assumptions on the generator of the time-change. This generalizes a result by Forni and Ulcigrai (JMD, 2012) for smooth time-changes of horocycle flows on compact surfaces., Comment: 11 pages

Let $G$ be a classical group with natural module $V$ and Lie algebra $\mathfrak{g}$ over an algebraically closed field $K$ of good characteristic. For rational irreducible representations $f: G \rightarrow \operatorname{GL}(W)$ occurring as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, we describe the Jordan normal form of $\mathrm{d} f(e)$ for all nilpotent elements $e \in \mathfrak{g}$. The description is given in terms of the Jordan block sizes of the action of $e$ on $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, for which recursive formulae are known. Our results are in analogue to earlier work (Proc. Amer. Math. Soc., 147 (2019) 4205-4219), where we considered these same representations and described the Jordan normal form of $f(u)$ for every unipotent element $u \in G$., to appear in J. Pure Appl. Algebra

Let $G$ be a connected reductive group defined over a non-archimedean local field $F$. Let $B$ be a minimal $F$-parabolic subgroup with Levi factor $T$ and unipotent radical $U$. Let $\psi$ be a non-degenerate character of $U(F)$ and $\lambda$ a character of $T(F)$. Let $(K,\rho)$ be a Bushnell-Kutzko type associated to the Bernstein block of $G(F)$ determined by the pair $(T,\lambda)$. We study the $\rho$-isotypical component $(c\text{-ind}_{U(F)}^{G(F)}\psi)^{\rho}$ of the induced space $c\text{-ind}_{U(F)}^{G(F)}\psi$ of functions compactly supported mod $U(F)$. We show that $(c\text{-ind}_{U(F)}^{G(F)}\psi)^{\rho}$ is cyclic module for the Hecke algebra $\mathcal{H}(G,\rho)$ associated to the pair $(K,\rho)$. When $T$ is split, we describe it more explicitly in terms of $\mathcal{H}(G,\rho)$. We make assumptions on the residue characteristic of $F$ and later also on the characteristic of $F$ and the center of $G$ depending on the pair $(T,\lambda)$. Our results generalize the main result of Chan and Savin in \cite{CS18} who treated the case of $\lambda=1$ for $T$ split., Comment: Statement of Theorem 3 corrected. To appear in Proceedings of the American Mathematical Society

In his theory of unipotent characters of finite groups of Lie type, Lusztig constructed modular categories from two-sided cells in Weyl groups. Brou\'e,Malle and Michel have extended parts of Lusztig's theory to complex reflection groups. This includes generalizations of the corresponding fusion algebras, although the presence of negative structure constants prevents them from arising from modular categories. We give here the first construction of braided pivotal monoidal categories associated with non-real reflection groups (later reinterpreted by Lacabanne as super modular categories). They are associated with cyclic groups, and their fusion algebras are those constructed by Malle., Comment: 26 pages

We introduce and study the maximal unipotent finite quotient for algebraic group schemes in positive characteristics. Applied to Picard schemes, this quotient encodes unusual torsion. We construct integral Fano threefolds where such unusual torsion actually appears. The existence of such threefolds is surprising, because the torsion vanishes for del Pezzo surfaces. Our construction relies on the theory of exceptional Enriques surfaces, as developed by Ekedahl and Shepherd-Barron., Comment: 29 pages; minor changes

In this paper we study the Gan-Gross-Prasad problem for finite classical groups. Our results provide complete answers for unipotent representations, and we obtain the explicit branching laws for these representations. Moreover, for arbitrary representations, we give a formula to reduce the Gan-Gross-Prasad problem to the restriction problem of Deligne-Lusztig characters.

Let $G$ be a semisimple real algebraic Lie group of real rank at least two and $U$ be the unipotent radical of a non-trivial parabolic subgroup. We prove that a discrete Zariski dense subgroup of $G$ that contains an irreducible lattice of $U$ is an arithmetic lattice of $G$. This solves a conjecture of Margulis and extends previous work of Hee Oh., Comment: 54 pages

Suppose that $G$ is a simple adjoint reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). Then there is a notion of modular forms for $G$, anchored on the so-called quaternionic discrete series representations of $G(\mathbf{R})$. The purpose of this paper is to give an explicit form of the Fourier expansion of modular forms on $G$, along the unipotent radical $N$ of the Heisenberg parabolic $P = MN$ of $G$., Comment: changed title; broadened definition of modular form; added discussion of constant term and Klingen Eisenstein series

Given a toric affine algebraic variety $X$ and a collection of one-parameter unipotent subgroups $U_1,\ldots,U_s$ of $\mathop{\rm Aut}(X)$ which are normalized by the torus acting on $X$, we show that the group $G$ generated by $U_1,\ldots,U_s$ verifies the following alternative of Tits' type: either $G$ is a unipotent algebraic group, or it contains a non-abelian free subgroup. We deduce that if $G$ is $2$-transitive on a $G$-orbit in $X$, then $G$ contains a non-abelian free subgroup, and so, is of exponential growth., 24 pages. The main result strengthened, the proof of Proposition 4.8 written in more detail; some references added; the referee remarks taken into account; the title changed

Let $G(q)$ be a Chevalley group over a finite field $F_q$. By Lusztig's and Shoji's work, the problem of computing the values of the unipotent characters of $G(q)$ is solved, in principle, by the theory of character sheaves; one issue in this solution is the determination of certain scalars relating two types of class functions on $G(q)$. We show that this issue can be reduced to the case where $q$ is a prime, which opens the way to use computer algebra methods. Here, and in a sequel to this article, we use this approach to solve a number of cases in groups of exceptional type which seemed hitherto out of reach., 20 pages; minor corrections and additions

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., Comment: arXiv admin note: text overlap with arXiv:1611.07373

Let $$\mathrm{F}$$ (resp. $$\mathbb F$$ ) be a nonarchimedean locally compact field with residue characteristic p (resp. a finite field with characteristic p). For $$k=\mathrm{F}$$ or $$k=\mathbb F$$ , let $$\mathbf {G}$$ be a connected reductive group over k and R be a commutative ring. We denote by $$\mathrm{Rep}( \mathbf G(k)) $$ the category of smooth R-representations of $$ \mathbf G(k) $$ . To a parabolic k-subgroup $${\mathbf P}=\mathbf {MN}$$ of $$\mathbf G$$ corresponds the parabolic induction functor $$\mathrm{Ind}_{\mathbf P(k)}^{\mathbf G(k)}:\mathrm{Rep}( \mathbf M(k)) \rightarrow \mathrm{Rep}( \mathbf G(k))$$ . This functor has a left and a right adjoint. Let $${{\mathcal {U}}}$$ (resp. $${\mathbb {U}}$$ ) be a pro-p Iwahori (resp. a p-Sylow) subgroup of $$ \mathbf G(k) $$ compatible with $${\mathbf P}(k)$$ when $$k=\mathrm{F}$$ (resp. $$\mathbb F$$ ). Let $${H_{ \mathbf G(k)}}$$ denote the pro-p Iwahori (resp. unipotent) Hecke algebra of $$ \mathbf G(k) $$ over R and $$\mathrm{Mod}({H_{ \mathbf G(k)}})$$ the category of right modules over $${H_{ \mathbf G(k)}}$$ . There is a functor $$\mathrm{Ind}_{{H_{ \mathbf M(k)}}}^{{H_{ \mathbf G(k)}}}: \mathrm{Mod}({H_{ \mathbf M(k)}}) \rightarrow \mathrm{Mod}({H_{ \mathbf G(k) }})$$ called parabolic induction for Hecke modules; it has a left and a right adjoint. We prove that the pro-p Iwahori (resp. unipotent) invariant functors commute with the parabolic induction functors, namely that $$\mathrm{Ind}_{\mathbf P(k)}^{\mathbf G(k)}$$ and $$\mathrm{Ind}_{{H_{ \mathbf M(k)}}}^{{H_{ \mathbf G(k)}}}$$ form a commutative diagram with the $${{\mathcal {U}}}$$ and $${{\mathcal {U}}}\cap \mathbf M(\mathrm{F})$$ (resp. $${\mathbb {U}}$$ and $${\mathbb {U}}\cap \mathbf M(\mathbb F) $$ ) invariant functors. We prove that the pro-p Iwahori (resp. unipotent) invariant functors also commute with the right adjoints of the parabolic induction functors. However, they do not commute with the left adjoints of the parabolic induction functors in general; they do if p is invertible in R. When R is an algebraically closed field of characteristic p, we show that an irreducible admissible R-representation of $$ \mathbf G(\mathrm{F}) $$ is supercuspidal (or equivalently supersingular) if and only if the $${H_{ \mathbf G(\mathrm{F})}}$$ -module $${\mathfrak {m}}$$ of its $${{\mathcal {U}}}$$ -invariants admits a supersingular subquotient, if and only if $${\mathfrak {m}}$$ is supersingular.

Let $\pi$ be an irreducible cuspidal representation of $\mathrm{GL}_{kn}\left(\mathbb{F}_q\right)$. Assume that $\pi = \pi_{\theta}$, corresponds to a regular character $\theta$ of $\mathbb{F}_{q^{kn}}^{*}$. We consider the twisted Jacquet module of $\pi$ with respect to a non-degenerate character of the unipotent radical corresponding to the partition $(n^k)$ of $kn$. We show that, as a $\mathrm{GL}_{n}\left(\mathbb{F}_q\right)$-representation, this Jacquet module is isomorphic to $\pi_{\theta \upharpoonright_{\mathbb{F}_n^*}} \otimes \mathrm{St}^{k-1}$, where $\mathrm{St}$ is the Steinberg representation of $\mathrm{GL}_{n}\left(\mathbb{F}_q\right)$. This generalizes a theorem of D. Prasad, who considered the case $k=2$. We prove and rely heavily on a formidable identity involving $q$-hypergeometric series and linear algebra., Comment: 27 pages. Comments are welcome

This contribution to the 2015 AMS Summer Institute in Algebraic Geometry (Salt Lake City) announces a general mirror construction. This construction applies to log Calabi-Yau pairs (X,D) with maximal boundary D or to maximally unipotent degenerations of Calabi-Yau manifolds. The new ingredient is a notion of "punctured Gromov-Witten invariant", currently in progress with Abramovich and Chen. The mirror to a pair (X,D) is constructed as the spectrum of a ring defined using the punctured invariants of (X,D). An analogous construction leads to mirrors of Calabi-Yau manifolds. This can be viewed as a generalization of constructions developed jointly with Hacking and Keel in the case of log CY surfaces and K3 surfaces., Comment: 37 pages, 1 figure. Current version corrects statement in Theorem 2.2: the algebra structure is not expected to be associative in complete generality, and some hypotheses are necessary. This does not affect the application to mirror symmetry constructions

We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let G be an algebraic group of classical type with defining characteristic p > 0 , μ a dominant weight and W the Weyl group of G. Let G = G ( q ) be a finite classical group, where q is a p-power. For a weight μ of G the sum s μ of distinct weights w ( μ ) with w ∈ W viewed as a function on the semisimple elements of G is known to be a generalized Brauer character of G called an orbit character of G. We compute, for certain orbit characters and every maximal torus T of G, the multiplicity of the trivial character 1 T of T in s μ . The main case is where μ = ( q − 1 ) ω and ω is a fundamental weight of G. Let St denote the Steinberg character of G. Then we determine the unipotent characters occurring as constituents of s μ ⋅ S t defined to be 0 at the p-singular elements of G. Let β μ denote the Brauer character of a representation of S L n ( q ) arising from an irreducible representation of G with highest weight μ. Then we determine the unipotent constituents of the characters β μ ⋅ S t for μ = ( q − 1 ) ω , and also for some other μ (called strongly q-restricted). In addition, for strongly restricted weights μ, we compute the multiplicity of 1 T in the restriction β μ | T for every maximal torus T of G.

A ring $R$ is a UU ring if every unit is unipotent, or equivalently if every unit is a sum of a nilpotent and an idempotent that commute. These rings have been investigated in C\u{a}lug\u{a}reanu \cite{C} and in Danchev and Lam \cite{DL}. In this paper, two generalizations of UU rings are discussed. We study rings for which every unit is a sum of a nilpotent and an idempotent, and rings for which every unit is a sum of a nilpotent and two idempotents that commute with one another.

We establish a fine expansion for the geometric part of the Arthur-Selberg trace formula (as it was conjectured by Werner Hoffmann). For the general linear group, we deduce an expression for the contributions of regular by blocks unipotent orbits (orbits with one Jordan block with multiplicity). As a consequence, we find formulas for Arthur's global coefficients attached to such orbits., Comment: in French

We prove a Lie-theoretic result for type A root systems. This allows us to determine the unipotent orbits attached to degenerate Eisenstein series on general linear groups, confirming a conjecture of David Ginzburg. In particular, this also shows that any unipotent orbit of general linear groups does occur as the unipotent orbit attached to a specific automorphic representation. The proof of the Lie-theoretic result relies on the notion of the descending decomposition, which expresses every Weyl group element as a product of simple reflections in a certain way. It is suitable for induction and allows us to translate the question into a combinatorial statement.

We prove a long-standing conjecture of Geck which predicts that cuspidal unipotent characters remain irreducible after $\ell$-reduction. To this end, we construct a progenerator for the category of representations of a finite reductive group coming from generalised Gelfand--Graev representations. This is achieved by showing that cuspidal representations appear in the head of generalised Gelfand--Graev representations attached to cuspidal unipotent classes, as defined and studied in \cite{GM96}., Comment: Version 2: Added a proof of Geck's conjecture, changed the title

Let $V$ be an $n$ dimensional vector space over an algebraic closure of a finite field $F_q$ and put $G = GL(V)$. For a positive integer $r$, we consider the variety $X_{uni} = G_{uni} \times V^{r-1}$, on which $G$ acts diagonally. $X_{uni}$ is the "unipotent part" of the enhanced variety of level $r$. $X_{uni}$ is partitioned into finitely many pieces $X_{\lambda}$ labelled by $r$-partitions $\lambda$ of $n$, and we consider the intersection cohomology $IC_{\lambda}$ associated to $X_{\lambda}$. In this paper, we show that the Frobenius trace functions (over $F_q$) associated to those $IC_{\lambda}$ satisfy certain orthogonality relations, which are very close to the equations characterizing the Kostka functions indexed by (a pair of) $r$-partitions. Using this we show, in some special cases, that the Kostka functions can be described in terms of those intersection cohomology, which is a (partial) generalization of the known results for the case $r = 1, 2$., Comment: v2: 52 pages. Proposition 6.3 was revised, and some examples were added in section 7. Final version, to appear in Transformation Groups

Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and linearization techniques, in view of Ratner’s measure classification for unipotent flows, reduce such problems to dynamical questions about linear actions of reductive groups on finite-dimensional vector spaces. This article provides general results which resolve these linear dynamical questions in terms of natural group theoretic or geometric conditions.

Let $G$ be a finitely generated torsion-free nilpotent group. The representation zeta function $\zeta_G(s)$ of $G$ enumerates twist isoclasses of finite-dimensional irreducible complex representations of $G$. We prove that $\zeta_G(s)$ has rational abscissa of convergence $a(G)$ and may be meromorphically continued to the left of $a(G)$ and that, on the line $\{s\in\mathbb{C} \mid \textrm{Re}(s) = a(G)\}$, the continued function is holomorphic except for a pole at $s=a(G)$. A Tauberian theorem yields a precise asymptotic result on the representation growth of $G$ in terms of the position and order of this pole. We obtain these results as a consequence of a more general result establishing uniform analytic properties of representation zeta functions of finitely generated nilpotent groups of the form $\mathbf{G}(\mathcal{O})$, where $\mathbf{G}$ is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal{O}$ of integers of a number field. This allows us to show, in particular, that the abscissae of convergence of the representation zeta functions of such groups and their pole orders are invariants of $\mathbf{G}$, independent of $\mathcal{O}$., Comment: 25 pages, final version, to appear in the Transactions of the American Mathematical Society

We show that the decomposition matrix of unipotent $\ell$-blocks of a finite reductive group $\mathbf{G}(\mathbb{F}_q)$ has a unitriangular shape, assuming $q$ is a power of a good prime and $\ell$ is very good for $\mathbf{G}$. This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand--Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka., Comment: v1: 71 pages. v2: 73 pages. Added an acknowledgment, some references, and Lemma 10.2. Minor linguistic changes throughout. v3: 74 pages. Post referee. Added Lem 6.9. Improved the assumtpions in Thm A. Updated the proof in Section 11. To appear Ann. of Math

Let K be a non-archimedean local field and let G be a connected reductive K-group which splits over an unramified extension of K. We investigate supercuspidal unipotent representations of the group G(K). We establish a bijection between the set of irreducible G(K)-representations of this kind and the set of cuspidal enhanced L-parameters for G(K), which are trivial on the inertia subgroup of the Weil group of K. The bijection is characterized by a few simple equivariance properties and a comparison of formal degrees of representations with adjoint $\gamma$-factors of L-parameters. This can be regarded as a local Langlands correspondence for all supercuspidal unipotent representations. We count the ensueing L-packets, in terms of data from the affine Dynkin diagram of G. Finally, we prove that our bijection satisfies the conjecture of Hiraga, Ichino and Ikeda about the formal degrees of the representations., Comment: Version 2: minor corrections and additions, mainly in section 15

Let $G$ and $\tilde G$ be reductive groups over a local field $F$. Let $\eta : \tilde G \to G$ be a $F$-homomorphism with commutative kernel and commutative cokernel. We investigate the pullbacks of irreducible admissible $G$-representations $\pi$ along $\eta$. Following Borel, Adler--Korman and Xu, we pose a conjecture on the decomposition of the pullback $\eta^* \pi$. It is formulated in terms of enhanced Langlands parameters and includes multiplicities. This can be regarded as a functoriality property of the local Langlands correspondence. We prove this conjecture for three classes: principal series representation of split groups (over non-archimedean local fields), unipotent representations (also with $F$ non-archimedean) and inner twists of $GL_n, SL_n, PGL_n$. Our main techniques involve Hecke algebras associated to Langlands parameters. We also prove a version of the pullback/functoriality conjecture for those., Comment: Version 2: corrections in 2.7, 2.8, 2.9. Paragraph 8.3 was abridged

We study a coarse moduli space of irreducible representations of the group of unipotent matrices of order $\mathbb{4}$ over the ring of integers which have finite weight. All such representations are known to be monomial. To describe a coarse moduli space of such representations, we need to study pairs of subgroups and their characters, which induce non-isomorphic irreducible representations. We obtain a full classification of such pairs and, respectively, a coarse moduli space., Comment: 38 pages