We study the Yangian $Y_n$ associated to the general linear Lie algebra $\mathfrak{gl}_n$ over a field of positive characteristic, as well as its shifted analog $Y_n(\sigma)$. Our main result gives a description of the centre of $Y_n(\sigma)$: it is a polynomial algebra generated by its {\em Harish-Chandra centre} (which lifts the centre in characteristic zero) together with a large {\em $p$-centre}. Moreover, $Y_n(\sigma)$ is free as a module over its center. In future work, it will be seen that every reduced enveloping algebra $U_\chi(\mathfrak{gl}_n)$ is Morita equivalent to a quotient of an appropriate choice of shifted Yangian, and so our results will have applications in classical representation theory.
We revisit the definition of the Heisenberg category of level $k \in \mathbb{Z}$. In level $- 1$, this category was introduced originally by Khovanov, but with some additional cyclicity relations which we show here are unnecessary. In other negative levels, the definition is due to Mackaay and Savage, also with some redundant relations, while the level zero case is the affine oriented Brauer category of Brundan, Comes, Nash and Reynolds. We also discuss cyclotomic quotients.
Let $W_{m|n}$ be the (finite) $W$-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb{C})$. In this paper we study the {\em Whittaker coinvariants functor}, which is an exact functor from category $\mathcal O$ for $\mathfrak{gl}_{m|n}(\mathbb{C})$ to a certain category of finite-dimensional modules over $W_{m|n}$. We show that this functor has properties similar to Soergel's functor $\mathbb{V}$ in the setting of category $\mathcal O$ for a semisimple Lie algebra. We also use it to compute the center of $W_{m|n}$ explicitly, and deduce some consequences for the classification of blocks of $\mathcal O$ up to Morita/derived equivalence.
We show that the blocks of category $\mathcal{O}$ for the Lie superalgebra $\mathfrak{q}_n(\mathbb{C})$ associated to half-integral weights carry the structure of a tensor product categorification for the infinite rank Kac-Moody algebra of type C$_\infty$. This allows us to prove two conjectures formulated by Cheng, Kwon and Lam. We then focus on the full subcategory consisting of finite-dimensional representations, which we show is a highest weight category with blocks that are Morita equivalent to certain generalized Khovanov arc algebras.
We introduce generalizations of Kac-Moody 2-categories in which the quiver Hecke algebras of Khovanov, Lauda and Rouquier are replaced by the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka.
We give a new proof of the "super Kazhdan-Lusztig conjecture" for the Lie superalgebra $\mathfrak{gl}_{n|m}(\mathbb{C})$ as formulated originally by the first author. We also prove for the first time that any integral block of category $\mathcal O$ for $\mathfrak{gl}_{n|m}(\mathbb{C})$ (and also all of its parabolic analogs) possesses a graded version which is Koszul. Our approach depends crucially on an application of the uniqueness of tensor product categorifications established recently by the second two authors.
This work is a companion to our article "Super Kac-Moody 2-categories," which introduces super analogs of the Kac-Moody 2-categories of Khovanov-Lauda and Rouquier. In the case of $\mathfrak{sl}_2$, the super Kac-Moody 2-category was constructed already in [A. Ellis and A. Lauda, "An odd categorification of $U_q(\mathfrak{sl}_2)$"], but we found that the formalism adopted there became too cumbersome in the general case. Instead, it is better to work with 2-supercategories (roughly, 2-categories enriched in vector superspaces). Then the Ellis-Lauda 2-category, which we call here a $\Pi$-2-category (roughly, a 2-category equipped with a distinguished involution in its Drinfeld center), can be recovered by taking the superadditive envelope then passing to the underlying 2-category. The main goal of this article is to develop this language and the related formal constructions, in the hope that these foundations may prove useful in other contexts.
The affine oriented Brauer category is a monoidal category obtained from the oriented Brauer category ($=$ the free symmetric monoidal category generated by a single object and its dual) by adjoining a polynomial generator subject to appropriate relations. In this article, we prove a basis theorem for the morphism spaces in this category, as well as for all of its cyclotomic quotients.
This is an expository article developing some aspects of the theory of categorical actions of Kac-Moody algebras in the spirit of works of Chuang-Rouquier, Khovanov-Lauda, Webster, and many others.
We show that every block of category $\mathcal{O}$ for the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb{C})$ is equivalent to some corresponding block of category $\mathcal{O}$ for the queer Lie superalgebra $\mathfrak{q}_{m+n}(\mathbb{C})$. This implies the truth of the Kazhdan-Lusztig conjecture for the so-called type A blocks of category $\mathcal{O}$ for the queer Lie superalgebra as formulated by Cheng, Kwon and Wang.
We show that the Kac-Moody 2-categories defined by Rouquier and by Khovanov and Lauda are the same.
This is a survey of some recent developments in the highest weight repesentation theory of the general linear Lie superalgebra $\mathfrak{gl}_{n|m}(\mathbb{C})$. The main focus is on the analog of the Kazhdan-Lusztig conjecture as formulated by the author in 2002, which was finally proved in 2011 by Cheng, Lam and Wang. Recently another proof has been obtained by the author joint with Losev and Webster, by a method which leads moreover to the construction of a Koszul-graded lift of category $\mathcal O$ for this Lie superalgebra.
This is a brief introduction to the quiver Hecke algebras of Khovanov, Lauda and Rouquier, emphasizing their application to the categorification of quantum groups. The text is based on lectures given by the author at the ICRA workshop in Bielefeld in August, 2012.
We give an algebraic construction of standard modules--infinite dimensional modules categorifying the PBW basis of the underlying quantized enveloping algebra--for Khovanov-Lauda-Rouquier algebras in all finite types. This allows us to prove in an elementary way that these algebras satisfy the homological properties of an "affine quasi-hereditary algebra." In simply-laced types these properties were established originally by Kato via a geometric approach. We also construct some Koszul-like projective resolutions of standard modules corresponding to multiplicity-free positive roots.
We consider the (finite) $W$-algebra $W_{m|n}$ attached to the principal nilpotent orbit in the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb{C})$. Our main result gives an explicit description of $W_{m|n}$ as a certain truncation of a shifted version of the Yangian $Y(\mathfrak{gl}_{1|1})$. We also show that $W_{m|n}$ admits a triangular decomposition and construct its irreducible representations.
We introduce some $\mathbb{Z}$-graded versions of the walled Brauer algebra $B_{r,s}(\delta)$, working over a field of characteristic zero. This allows us to prove that $B_{r,s}(\delta)$ is Morita equivalent to an idempotent truncation of a certain infinite dimensional version of Khovanov's arc algebra. We deduce that the walled Brauer algebra is Koszul whenever $\delta \neq 0$.
This is a survey of some recent results relating Khovanov's arc algebra to category $\mathcal O$ for Grassmannians, the general linear supergroup, and the walled Brauer algebra. The exposition emphasizes an extension of Young's orthogonal form for level two cyclotomic Hecke algebras.
We prove that blocks of the general linear supergroup are Morita equivalent to a limiting version of Khovanov's diagram algebra. We deduce that blocks of the general linear supergroup are Koszul.
This is the first of four articles studying some slight generalisations $H^n_m$ of Khovanov's diagram algebra, as well as quasi-hereditary covers $K^n_m$ of these algebras in the sense of Rouquier, and certain infinite dimensional limiting versions $K^\infty_m$, $K^{\pm \infty}_m$ and $K^\infty_\infty$. In this article we prove that $H^n_m$ is a cellular symmetric algebra and that $K^n_m$ is a cellular quasi-hereditary algebra. In subsequent articles, we relate $H^n_m, K^n_m$ and $K^\infty_m$ to level two blocks of degenerate cyclotomic Hecke algebras, parabolic category $\mathcal O$ and the general linear supergroup, respectively.
We use the theory of finite $W$-algebras associated to nilpotent orbits in the Lie algebra $\mathfrak{g} = \mathfrak{gl}_N(\mathbb{C})$ to give another proof of Moeglin's theorem about completely prime primitive ideals in the enveloping algebra $U(\mathfrak{g})$. We also make some new observations about Joseph's Goldie rank polynomials in Cartan type $A$.
We give a presentation for the cohomology algebra of the Spaltenstein variety of all partial flags annihilated by a fixed nilpotent matrix, generalizing the description of the cohomology algebra of the Springer fiber found by De Concini, Procesi and Tanisaki.
Recently, the first two authors have defined a $\mathbb Z$-grading on group algebras of symmetric groups and more generally on the cyclotomic Hecke algebras of type $G(l,1,d)$. In this paper we explain how to grade Specht modules over these algebras.
We prove that integral blocks of parabolic category $\mathcal O$ associated to the subalgebra $\mathfrak{gl}_m(\mathbb{C}) \oplus \mathfrak{gl}_n(\mathbb{C})$ of $\mathfrak{gl}_{m+n}(\mathbb{C})$ are Morita equivalent to quasi-hereditary covers of generalised Khovanov algebras. Although this result is in principle known, the existing proof is quite indirect, going via perverse sheaves on Grassmannians. Our new approach is completely algebraic, exploiting Schur-Weyl duality for higher levels. As a by-product we get a concrete combinatorial construction of $2$-Kac-Moody representations in the sense of Rouquier corresponding to level two weights in finite type $A$.
This is the second of a series of four articles studying various generalisations of Khovanov's diagram algebra. In this article we develop the general theory of Khovanov's diagrammatically defined "projective functors" in our setting. As an application, we give a direct proof of the fact that the quasi-hereditary covers of generalised Khovanov algebras are Koszul.
We explain how to deduce the degenerate analogue of Ariki's categorification theorem over the ground field $\mathbb C$ as an application of Schur-Weyl duality for higher levels and the Kazhdan-Lusztig conjecture in finite type A. We also discuss some supplementary topics, including Young modules, tensoring with sign, tilting modules and Ringel duality.
In recent joint work with Wang, we have constructed graded Specht modules for cyclotomic Hecke algebras. In this article, we prove a graded version of the Lascoux-Leclerc-Thibon conjecture, describing the decomposition numbers of graded Specht modules over a field of characteristic zero.
We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) cyclotomic Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki's categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial $\mathbb{Z}$-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.
We construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the general linear Lie algebra over a field of characteristic zero. In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova.
We study highest weight representations of shifted Yangians over an algebraically closed field of characteristic 0. In particular, we classify the finite dimensional irreducible representations and explain how to compute their Gelfand-Tsetlin characters in terms of known characters of standard modules and certain Kazhdan-Lusztig polynomials. Our approach exploits the relationship between shifted Yangians and the finite W-algebras associated to nilpotent orbits in general linear Lie algebras.
We extend Schur-Weyl duality to an arbitrary level $l \geq 1$, level one recovering the classical duality between the symmetric and general linear groups. In general, the symmetric group is replaced by the degenerate cyclotomic Hecke algebra over $\mathbb{C}$ parametrized by a dominant weight of level $l$ for the root system of type $A_\infty$. As an application, we prove that the degenerate analogue of the quasi-hereditary cover of the cyclotomic Hecke algebra constructed by Dipper, James and Mathas is Morita equivalent to certain blocks of parabolic category $\mathcal{O}$ for the general linear Lie algebra.
We define analogues of Verma modules for finite W-algebras. By the usual ideas of highest weight theory, this is a first step towards the classification of finite dimensional irreducible modules. We also introduce an analogue of the BGG category $\mathcal O$. Motivated by known results in type A, we then formulate some precise conjectures in the case of nilpotent orbits of standard Levi type.
We prove that the center of a regular block of parabolic category $\mathcal O$ for the general linear Lie algebra is isomorphic to the cohomology algebra of a corresponding Springer fiber. This was conjectured by Khovanov. We also find presentations for the centers of singular blocks, which are cohomology algebras of Spaltenstein varieties.
We prove that the center of each degenerate cyclotomic Hecke algebra associated to the complex reflection group of type $B_d(l)$ consists of symmetric polynomials in its commuting generators. The classification of the blocks of the degenerate cyclotomic Hecke algebras is an easy consequence. We then apply Schur-Weyl duality for higher levels to deduce analogous results for parabolic category $\mathcal O$ for the Lie algebra $\mathfrak{gl}_n(\mathbb{C})$.
Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over $\mathbb{C}$ and $e \in \mathfrak{g}$ a nilpotent element. Elashvili and Kac have recently classified all {good $\mathbb{Z}$-gradings} for $e$. We instead consider {\em good $\mathbb{R}$-gradings}, which are naturally parameterized by an open convex polytope in a Euclidean space arising from the reductive part of the centralizer of $e$ in $\mathfrak{g}$. As an application, we prove that the isomorphism type of the finite W-algebra attached to a good $\mathbb{R}$-grading for $e$ is independent of the particular choice of good grading.
We derive a formula for the entries of the (unitriangular) transition matrices between the standard monomial and dual canonical bases of the irreducible polynomial representations of $U_q(\mathfrak{gl}_n)$ in terms of Kazhdan-Lusztig polynomials.
This paper is concerned with the modular representation theory of the affine Hecke-Clifford superalgebra, the cyclotomic Hecke-Clifford superalgebras, and projective representations of the symmetric group. Our approach exploits crystal graphs of affine Kac-Moody algebras.
We give a presentation for the finite W-algebra associated to a nilpotent matrix in the general linear Lie algebra over $\mathbb{C}$. In the special case that the nilpotent matrix consists of $n$ Jordan blocks each of the same size $l$, the presentation is that of the Yangian of level $l$ associated to $\mathfrak{gl}_n$, as was first observed by Ragoucy and Sorba. In the general case, we are lead to introduce some generalizations of the Yangian which we call the shifted Yangians.
We continue our study of the representations of the supergroup $Q(n)$ over a field of odd positive characteristic. The focus here is on the aspects of the theory that depend in some way on the interpretation of induction in terms of sheaf cohomology of certain equivariant vector bundles on the associated flag superschemes.
We introduce some new presentations for the Yangian associated to the Lie algebra $\mathfrak{gl}_n$. These presentations are parametrized by tuples of positive integers summing to $n$. At one extreme, for the tuple $(n)$, the presentation is the usual RTT presentation of $Y_n$. At the other extreme, for the tuple $(1^n)$, the presentation is closely related to Drinfeld's presentation. In general, the presentations are useful for understanding the structure of the standard parabolic subalgebras of $Y_n$.
We develop a general theory of tilting modules for graded Lie superalgebras, extending work of Soergel for graded Lie algebras. The main result of the article gives a twisted version of BGG reciprocity relating multiplicities in $\Delta$-flags of indecomposable tilting modules to composition multiplicities of costandard modules. We then discuss the examples $\mathfrak{gl}(m|n)$ and $\mathfrak{q}(n)$ in detail.
We compute the characters of the finite dimensional irreducible representations of the Lie superalgebra $\mathfrak{q}(n)$ and formulate a precise conjecture for the other irreducible characters in category $\mathcal O$. The guiding principle is that the representation theory of $\mathfrak q(n)$ should be described by Lusztig's canonical basis for "tensor space" over the quantum group of type $\mathfrak b_\infty$.
We present a new proof of the Mullineux conjecture, which describes the effect of tensoring an irreducible modular representation of the symmetric group with the sign representation. Our approach exploits the "super" Schur-Weyl duality connecting representations of the symmetric group to that of the general linear supergroup $GL(n|n)$. We also give for the first time a classification of the irreducible polynomial representations of $GL(m|n)$ in positive characteristic.
In this survey article, we give an overview of the new Lie theoretic approach to the $p$-modular representation theory of the symmetric groups and their double covers that has emerged in the last few years.
We compute the characters of the finite dimensional irreducible representations of the Lie superalgebra $\mathfrak{gl}(m|n)$, and determine $\operatorname{Ext}$'s between simple modules in the category of finite dimensional representations. We formulate conjectures for the analogous results in category $\mathcal O$. The combinatorics parallels the combinatorics of certain canonical bases over the Lie algebra $\mathfrak{gl}(\infty)$.
We investigate modular representations of the supergroup $Q(n)$ in positive characteristic, motivated by connections to the modular representation theory of the spin symmetric groups. In particular, we classify the irreducible representations and prove a version of the Steinberg tensor product theorem.
We compute the determinant of the Gram matrix of the Shapovalov form on weight spaces of the basic representation of an affine Kac-Moody algebra of ADE type (possibly twisted). As a consequence, we obtain explicit formulae for the determinants of the Cartan matrices of $p$-blocks of the symmetric group and its double cover, and of the associated Hecke algebras at roots of unity.
We exploit the "super" Schur-Weyl duality between the spin symmetric groups and the supergroup $Q(n)$ to establish for the first time a classification of the irreducible modular representations of the spin symmetric groups.
We give a formula for the values of irreducible unipotent $p$-modular Brauer characters of $GL_n(\mathbb{F}_q)$ at unipotent elements, where $p$ is a prime not dividing $q$, in terms of (unknown!) weight multiplicities of quantum $GL_n$ and certain generic polynomials $S_{\lambda,\mu}(q)$. These polynomials arise as entries of the transition matrix between the renormalized Hall-Littlewood symmetric functions and the forgotten symmetric functions. We also provide an alternative combinatorial algorithm working in the Hall algebra for computing $S_{\lambda,\mu}(q)$.
This paper is concerned with the modular representation theory of the affine Hecke-Clifford superalgebra, the cyclotomic Hecke-Clifford superalgebras, and projective representations of the symmetric group. Our approach exploits crystal graphs of affine Kac-Moody algebras.
Note this version contains some additional tables in the appendix that were not in the published version!
We give a self-contained account of the results originating in the work of James and the second author in the 1980s relating the representation theory of $GL_n(\mathbb{F}_q)$ over fields of characteristic coprime to $p$ to the representation theory of "quantum $GL_n$" at roots of unity.
The new treatment allows us to extend the theory in several directions. First, we prove a precise functorial connection between the operations of tensor product in quantum $GL_n$ and Harish-Chandra induction in finite $GL_n$. This allows us to obtain a version of the recent Morita theorem of Cline, Parshall and Scott valid in addition for $p$-singular classes.
From that we obtain simplified treatments of various basic known facts, such as the computation of decomposition numbers and blocks of $GL_n(\mathbb{F}_q)$ from knowledge of the same for the quantum group, and the non-defining analogue of Steinberg's tensor product theorem. We also easily obtain a new double centralizer property between $GL_n(\mathbb{F}_q)$ and quantum $GL_n$ generalizing a result of Takeuchi.
Finally, we apply the theory to study the affine general linear group, following ideas of Zelevinsky in characteristic zero. We prove results that can be regarded as the modular analogues of Zelevinsky's and Thoma's branching rules. Using these, we obtain a new dimension formula for the irreducible cross-characteristic representations of $GL_n(\mathbb{F}_q)$, expressing their dimensions in terms of the characters of irreducible modules over the quantum group.
Let $F$ be an algebraically closed field of characteristic $p$, and $\Sigma_n$ be the symmetric group on $n$ letters. In this paper we classify all pairs $(G,D)$, where $D$ is an irreducible $F\Sigma_n$-module of dimension greater than $1$ and $G$ is a proper subgroup of $\Sigma_n$, such that the restriction $D{\downarrow}_G$ is irreducible, provided $p>3$.
The goal of this article is to give an exposition of some recent results on tensor products and restrictions for rational representations of the general linear group in positive characteristic. We also outline the relations with the LLT algorithm and the ideal structure of the group algebra of the finitary symmetric group.
Let $G$ denote the finite general linear group $\operatorname{GL}_n(\mathbb{F}_q)$ over the finite field with $q$ elements. Associated to each partition $\lambda$ of $n$, there is an irreducible unipotent complex character $\chi_\lambda$ of $G$. The degree of $\chi_\lambda$ is a polynomial in $q$ given by Green's hook formula; the polynomial is monic of degree $b(\lambda')$ where $\lambda'$ is the transpose of $\lambda$ and, for a partition $\mu = (m_1 \geq m_2 \geq \dots \geq m_h > 0)$ of $n$, $b(\mu)$ denotes $\frac{n(n+1)}{2} - \sum_{i=1}^h i m_i$. An easy consequence of Green's formula is that $ \chi_\lambda(1) \geq q^{b(\lambda')}. $ The purpose of this note is to prove similar lower bounds for the degrees of the irreducible $p$-modular Brauer characters of $G$ when $p$ is coprime to $q$.
We classify all pairs of reductive maximal connected subgroups of a classical algebraic group $G$ that have a dense double coset in $G$. Using this, we show that for an arbitrary pair $(H, K)$ of reductive subgroups of a reductive group $G$ satisfying a certain mild technical condition, there is a dense $H, K$-double coset in $G$ precisely when $G = HK$ is a factorization.
Let $\mathbb{F}$ be an algebraically closed field of characteristic $p$. A central problem in the representation theory of the general linear group $GL(n) = GL_n(\mathbb{F})$ is to understand the structure of the tensor space $ V^{\otimes r}, $ where $V$ is the natural $GL(n)$-module. To do this (inductively) we would like information about the structure of tensor products of the form $M \otimes V$, where $M$ is an irreducible (or Weyl or tilting) module. Given $\alpha \in \mathbb{Z} / p\mathbb{Z}$, we will define functors $Tr^\alpha$ and $Tr_\alpha$, which roughly speaking are given by tensoring with the natural $GL(n)$-module $V$ and its dual $V^*$ respectively, then projecting onto certain blocks determined by the residue $\alpha$. Our main results give some detailed structural information about the effect of these functors on irreducible $GL(n)$-modules. We also deduce analogous results for modular representations of the symmetric group, where $Tr^\alpha$ and $Tr_\alpha$ correspond to the $\alpha$-induction and $\alpha$-restriction functors.
We prove four different results relating what could be called "modular Littlewood-Richardson coefficients" for the general linear and symmetric groups.
In this paper we derive some relations between the branching rules from $GL_n(\mathbb{F})$ to its Levi subgroups on one hand, and decompositions of tensor products over $GL_n(\mathbb{F})$ itself on the other. Some of our results are valid for arbitrary reductive groups.
We prove the quantum analogue of Kleshchev's modular branching rules, i.e. we replace the symmetric and general linear groups in Kleshchev's results by the corresponding Iwahori-Hecke algebra and the quantised enveloping algebra $U_q(\mathfrak{gl}_n)$.
Let $G$ be a simple algebraic group of exceptional type, defined over an algebraically closed field $K$ of characteristic $p \geq 0$. In this paper, we classify all pairs $(X, Y)$ of reductive subgroups of $G$ which have a dense $(X, Y)$-double coset in $G$. In fact, we show that there is a dense $(X, Y)$-double coset in $G$ precisely when $G = XY$ is a factorisation.
Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $k$ of characteristic $p \geq 0$. We are mainly interested here in the case of positive characteristic. We will describe some results on the problem of finding subgroups $H$ and $P$ with a dense $H, P$-double coset in $G$.
We obtain a criterion for the restriction of an irreducible rational $GL(n)$-module to the naturally embedded subgroup $GL(n-1)$ to be semisimple, over an arbitrary algebraically closed field. In that case, we describe the composition factors of the restriction explicitly. As an application, we classify the completely splittable representations of general linear groups and give an exact character formula for these modules.
We describe some developments in the representation theory of $GL(n)$ which depend on certain {\em lowering operators} recently discovered by Kleshchev. We give a simple new definition of these lowering operators and explain the relationship between these and operators which have previously appeared in the work of Carter-Lusztig and others. Our approach simplifies two important applications: the construction of orthogonal bases for Weyl modules over $\mathbb{C}$ and Kleshchev's modular branching rules for symmetric groups. We also describe previously unknown analogues of these two results in the quantum case.
We introduce the notion of a multiplicity-free subgroup of a reductive algebraic group in arbitrary characteristic. This concept already exists in the work of Krämer for compact connected Lie groups. We give a classification of reductive multiplicity-free subgroups, and as a consequence obtain a simple proof of a theorem of Kleshchev.
Let $H$ be a closed reductive subgroup of a connected reductive algebraic group $G$. We prove by some case analysis that there is no dense $(H,H)$-double coset in $G$. (Note: a case-free proof is given in Proposition 1.6 of my subsequent article "Double coset density in exceptional algebraic groups.")