In this article we consider sheaf quotients of affine superschemes by affine supergroups that act on them freely. The necessary and sufficient conditions for such quotients to be affine are given. If $G$ is an affine supergroup and $H$ is its normal supersubgroup, then we prove that a dur $K$-sheaf $\tilde{\tilde{G/H}}$ is again affine supergroup. Additionally, if $G$ is algebraic, then a $K$-sheaf $\tilde{G/H}$ is also algebraic supergroup and it coincides with $\tilde{\tilde{G/H}}$. In particular, any normal supersubgroup of an affine supergroup is faithfully exact.
In this article we consider sheaf quotients of affine superschemes by finite supergroups that act on them freely. More precisely, if a finite supergroup $G$ acts on an affine superscheme $X$ freely, then the quotient $K$-sheaf $\tilde{X/G}$ is again an affine superscheme $Y$, where $K[Y]\simeq K[X]^G$. Besides, $K[X]$ is a finitely presented projective $K[X]^G$-module.
To generalize some fundamental results on group schemes to the super context, we study the quotient sheaf $G \tilde{/} H$ of an algebraic supergroup $G$ by its closed supersubgroup $H$, in arbitrary characteristic $\neq$ 2. Our main theorem states that $G \tilde{/} H$ is a Noetherian superscheme. This together with derived results give positive answers to interesting questions posed by J. Brundan.
Over an arbitrary field of characteristic $\ne 2$, we define the notion of Harish-Chandra pairs, and prove that the category of those pairs is anti-equivalent to the category of algebraic affine supergroup schemes. The result is applied to characterize some classes of affine supergroup schemes such as those which are (a) simply connected, (b) unipotent or (c) linearly reductive in positive characteristic.
This paper dwells upon two aspects of affine supergroup theory, investigating the links among them. First, I discuss the "splitting" properties of affine supergroups, i.e. special kinds of factorizations they may admit - either globally, or pointwise. Second, I present a new contribution to the study of affine supergroups by means of super Harish-Chandra pairs (a method already introduced by Koszul, and later extended by other authors). Namely, I provide an explicit, functorial construction Ψwhich, with each super Harish-Chandra pair, associates an affine supergroup that is always globally str…
We present two novel proofs of the known classification of connected affine algebraic supergroups $G$ such that $\operatorname{Rep}G$ is semisimple. The proofs are geometrically motivated, although both rely on an algebraic lemma that characterizes $\mathfrak{osp}(1|2n)$ amongst simple Lie superalgebras.
We prove that given a super affine closed subgroup $H$ of a super affine group $G$ over a field $k$ of charctersitic $\mathrm{ch} k \ne 2$, the dur $k$-sheaf $G\tilde{\tilde{/}} H$ of right cosets is affine if the affine $k$-group $\bar{H}$ assocoiated to $H$ is (a) reductive or (b) pro-finite. Especially when $G$ is algebraic, the result in Case (a) gives rise to a positive answer to Brundan's question which was recently discussed by Zubkov \cite{Z}.
We show that every connected affine algebraic supergroup defined over a field k, with diagonalizable maximal torus and whose tangent Lie superalgebra is a k-form of a complex simple Lie superalgebra of classical type is a Chevalley supergroup, as it is defined and constructed explicitly in [R. Fioresi, F. Gavarini, "Chevalley Supergroups", Memoirs of the AMS 215 (2012), no. 1014].
I present a construction of connected affine algebraic supergroups G_V associated with simple Lie superalgebras g of Cartan type and with g-modules V. Conversely, I prove that every connected affine algebraic supergroup whose tangent Lie superalgebra is of Cartan type is necessarily isomorphic to one of the supergroups G_V that I introduced. In particular, the supergroup constructed in this way associated with g := W(n) and its standard representation is described somewhat more in detail. In addition, *** an "Erratum" is added here *** after the main text to fix a mistake which was kindly poin…
The affine Kauffmann category is a strict monoidal category and can be considered as a $q$-analogue of the affine Brauer category in (Rui et al. in Math. Zeit. 293, 503-550, 2019). In this paper, we prove a basis theorem for the morphism spaces in the affine Kauffmann category. The cyclotomic Kauffmann category is a quotient category of the affine Kauffmann category. We also prove that any morphism space in this category is free over an integral domain $\mathbb K$ with maximal rank if and only if the $\mathbf u$-admissible condition holds in the sense of Definition 1.13.
In the framework of algebraic supergeometry, we give a construction of the scheme-theoretic supergeometric analogue of Chevalley groups, namely affine algebraic supergroups associated to simple Lie superalgebras of classical type. In particular both basic (even exceptional) and strange cases are covered. This provides a unified approach to most of the algebraic supergroups considered so far in literature, and an effective method to construct new ones. As an intermediate step, we prove an existence theorem for Chevalley bases of simple classical Lie superalgebras and a PBW-like theorem for thei…
After establishing a geometric Schur-Weyl duality in a general setting, we recall this duality in type A in the finite and affine case. We extend the duality in the affine case to positive parts of the affine algebras. The positive parts have nice ideals coming from geometry, allowing duality for quotients. Some of the quotients of the positive affine Hecke algebra are then identified to some cyclotomic Hecke algebras and the geometric setting allows the construction of canonical bases.
We study a class of representations over the degenerate double affine Hecke algebra of gl_n by an algebraic method. As fundamental objects in this class, we introduce certain induced modules and study some of their properties. In particular, it is shown that these induced modules have unique simple quotient modules under certain conditions. Moreover, we show that any simple module in this class is obtained as such a simple quotient, and give a classification of all the simple modules.
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.
A semisimple algebraic tensor category over an algebraically closed field k of characteristic zero is the representation category of all finite dimensional twisted super representations of an affine reductive supergroup G over k. Such a supergroup is reductive if and only if its connected component is reductive. The connected component is reductive if and only if the Lie superalgebra divided by its center is a product of simple Lie algebras of classical type and Lie superalgebras spo(1,2r) of the orthosymplectic types BC_r.
We define filtered ADHM data and connect a notion of filtered quiver representations to Grothendieck--Springer resolutions. We also provide current developments and give a list of research problems to further study filtered ADHM equation.
We study the Borel moment map $μ_B:T^*(\mathfrak{b}\times \mathbb{C}^n)\rightarrow \mathfrak{b}^*$, given by $(r,s,i,j)\mapsto [r,s]+ij$, and describe our algorithm to construct the geometric invariant theory (GIT) quotients $μ_B^{-1}(0)/\!\!/_{\det}B$ and $μ_B^{-1}(0)/\!\!/_{\det^{-1}}B$, and the affine quotient $μ_B^{-1}(0)/\!\!/B$. We also provide an insight of the singular locus of $2^n$ irreducible components of $μ_B$. Finally, analogous to the Hilbert--Chow morphism, we discuss that the GIT quotient for the Borel setting is a resolution of singularities.
We study products of the affine geometric crystal of type A corresponding to symmetric powers of the standard representation. The quotient of this product by the R-matrix action is constructed inside the unipotent loop group. This quotient crystal has a semi-infinite limit, where the crystal structure is described in terms of limit ratios previously appearing in the study of total positivity of loop groups.
For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies some simple condition.
We compute motivic Donaldson-Thomas invariants for crepant resolutions of quotients of affine three-space by even dihedral groups in terms of an affine type D root system, using double dimensional reduction and the representation theory of affine type D quivers.
Affine Weyl groups and their parabolic quotients are used extensively as indexing sets for objects in combinatorics, representation theory, algebraic geometry, and number theory. Moreover, in the classical Lie types we can conveniently realize the elements of these quotients via intuitive geometric and combinatorial models such as abaci, alcoves, coroot lattice points, core partitions, and bounded partitions. Berg, Jones, and Vazirani described a bijection between n-cores with first part equal to k and (n-1)-cores with first part less than or equal to k, and they interpret this bijection in te…
We prove that the quotient of the polynomial representation of the double affine Hecke algebra (DAHA) by the radical of the duality pairing is always irreducible assuming that it is finite dimensional (apart from the roots of unity). We also find necessary and sufficient conditions for the radical to be zero, which is a q-generalization of Opdam's formula for the singular k-parameters with the multiple zero-eigenvalues of the corresponding Dunkl operators.
The Hecke group algebra $HW_0$ of a finite Coxeter group $W_0$, as introduced by the first and last author, is obtained from $W_0$ by gluing appropriately its 0-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when $W_0$ is the classical Weyl group associated to an affine Weyl group $W$. Namely, we prove that, for $q$ not a root of unity, $HW_0$ is the natural quotient of the affine Hecke algebra through its level 0 representation. We further show that the level 0 representation is a calibrated principal series representation for a …
We study the geometry of the stratification induced by an affine hyperplane arrangement H on the quotient of a complex affine space by the action of a discrete group preserving H. We give conditions ensuring normality or normality in codimension 1 of strata. As an application, we provide the list of those categorical quotients of closures of Jordan classes and of sheets in all complex simple algebraic groups that are normal. In the simply connected case, we show that normality of such a quotient is equivalent to its smoothness.
We consider two linear reductive algebraic groups $ G $ and $ G' $ over $ C $. Take a finite dimensional rational representation $ W $ of $ G \times G' $. Let $ Y = W // G := Spec C[W]^G $ and $ X = W // G' := \Spec C[W]^{G'} $ be the affine quotients. The quotient space $ X $ (respectively $ Y $) naturally inherits the action of $ G $ (respectively $ G' $). In this note, we study the interrelation between the orbit structures of $ X / G $ and $ Y / G' $. In a good situation, we can embed $ Y / G' $ into $ X / G $, and the embedding map preserves important properties such as the closure relati…
The space spanned by the characters of twisted affine Lie algebras admit the action of certain congruence subgroups of $SL(2,\mathbb{Z})$. By embedding the characters in the space spanned by theta functions, we study an $SL(2,\mathbb{Z})$-closure of the space of characters. Analogous to the untwisted affine Lie algebra case, we construct a commutative associative algebra (fusion algebra) structure on this space through the use of the Verlinde formula and study important quotients. Unlike the untwisted cases, some of these algebras and their quotients, which relate to the trace of diagram autom…
Following Steinberg, we construct an adjoint quotient for the Vinberg semi-group and a section to this quotient. Then, after Ngô, we show the existence of a regular centralizer on it and use it to compute the affine Springer fibers for groups.
For a connected reductive group G and a finite-dimensional G-module V, we study the invariant Hilbert scheme that parameterizes closed G-stable subschemes of V affording a fixed, multiplicity-finite representation of G in their coordinate ring. We construct an action on this invariant Hilbert scheme of a maximal torus T of G, together with an open T-stable subscheme admitting a good quotient. The fibers of the quotient map classify affine G-schemes having a prescribed categorical quotient by a maximal unipotent subgroup of G. We show that V contains only finitely many multiplicity-free G-subva…
We introduce a notion of $ξ$-stability on the affine grassmannian $\xx$ for the classical groups, this is the local version of the $ξ$-stability on the moduli space of Higgs bundles on a curve introduced by Chaudouard and Laumon. We prove that the quotient $\xx^ξ/T$ of the stable part $\xx^ξ$ by the maximal torus $T$ exists as an ind-$k$-scheme, and we introduce a reduction process analogous to the Harder-Narasimhan reduction for vector bundles. For the group $\mathrm{SL}_{d}$, we calculate the Poincaré series of the quotient $\xx^ξ/T$.
The flag variety of a complex reductive linear algebraic group G is by definition the quotient G/B by a Borel subgroup. It can be regarded as the set of Borel subalgebras of Lie(G). Given a nilpotent element e in Lie(G), one calls Springer fiber the subvariety formed by the Borel subalgebras which contain e. Springer fibers have in general a complicated structure (not irreducible, singular). Nevertheless, a theorem by C. De Concini, G. Lusztig, and C. Procesi asserts that, when G is classical, a Springer fiber can always be paved by finitely many subvarieties isomorphic to affine spaces. In th…
In an unpublished preprint, A. King conjectured that there are tilting bundles over projective varieties which are obtained as invariant quotients of affine spaces for linear actions of reductive groups. The goal of this paper is to give further examples of strong exceptional sequences of vector bundles on certain projective varieties. These are obtained as geometric invariant quotients of affine spaces by linear actions of reductive groups, as in King's conjecture.
We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type $\tilde{C}$, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for both algebras. The elements of these bases are parameterized by certain subsets of fully commutative elements. We enumerate these elements according to their affine length.
