Complex Algebraic Group Research Articles

Construction of even-level representations of [formula omitted] with residue field of characteristic two

We construct the finite-dimensional continuous complex representations of SL2 over compact discrete valuation rings of even residual characteristic, assuming the level is large enough compared to the ramification index, in the mixed characteristic case. We also prove that the complex group algebras of SL2 over finite quotient rings of such compact discrete valuation rings depend on the characteristic of the ring. In particular, we prove that the group algebras C[SL2(Z/2rZ)] and C[SL2(F2[t]/(tr))] are not isomorphic for any r≥4.

Torsors on moduli spaces of principal G-bundles on curves

Let [Formula: see text] be a semisimple complex algebraic group with a simple Lie algebra [Formula: see text], and let [Formula: see text] denote the moduli stack of topologically trivial stable [Formula: see text]-bundles on a smooth projective curve [Formula: see text]. Fix a theta characteristic [Formula: see text] on [Formula: see text] which is even in case [Formula: see text] is odd. We show that there is a nonempty Zariski open substack [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text], for all [Formula: see text]. It is shown that any such [Formula: see text] has a canonical connection. It is also shown that the tangent bundle [Formula: see text] has a natural splitting, where [Formula: see text] is the restriction of [Formula: see text] to the semi-stable locus. We also produce an isomorphism between two naturally occurring [Formula: see text]-torsors on the moduli space of regularly stable [Formula: see text].

MV polytopes and reduced double Bruhat cells

When G is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of G . By fixing w from the Weyl group, we can define MV polytopes whose highest vertex is labelled by w . We show that these polytopes are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by w^{-1} . To do this, we define a collection of generalized minor functions \Delta_{\gamma}^{\mathrm{new}} which tropicalize on the reduced Bruhat cell to the BZ data of an MV polytope of highest vertex w . We also describe the combinatorial structure of MV polytopes of highest vertex w . We explicitly describe the map from the Weyl group to the subset of elements bounded by w in the Bruhat order which sends u \mapsto v if the vertex labelled by u coincides with the vertex labelled by v for every MV polytope of highest vertex w . As a consequence of this map, we prove that these polytopes have vertices labelled by Weyl group elements less than w in the Bruhat order.

Symplectic complexity of reductive group actions

Let a complex algebraic reductive group G act on a complex algebraic manifold X. For a G-invariant subvariety Ξ of the nilpotent cone N(g∗)⊂g∗ we define a notion of Ξ-symplectic complexity of X. This notion generalizes the notion of complexity defined in Vinberg (1986). We prove several properties of this notion, and relate it to the notion of Ξ-complexity defined in Aizenbud and Gourevitch (2024) motivated by its relation with representation theory.

On twisted group ring isomorphism problem for p-groups

Abstract In this article, we explore the problem of determining isomorphisms between the twisted complex group algebras of finite $p$ -groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently examined by Margolis-Schnabel. Our focus lies on a specific invariant, referred to as the generalized corank, which relates to the twisted complex group algebra isomorphism problem. We provide a solution for non-abelian $p$ -groups with generalized corank at most three.

Exploring projective equivalences between closures of orbits

Motivated by results in the literature that use representations and group actions to produce nice geometric results about algebraic varieties, this article studies projective equivalence relations between closures of orbits for several complex algebraic group actions on , where is a complex representation of . In particular, we study the cases when is one of the following:, , , and . On the way, we also obtain some interesting geometric results from studying these orbits.

Torus Actions on Quotients of Affine Spaces

We study the locus of fixed points of a torus action on a GIT quotient of a complex vector space by a reductive complex algebraic group which acts linearly. We show that, under the assumption that $G$ acts freely on the stable locus, the components of the fixed point locus are again GIT quotients of linear subspaces by Levi subgroups.

Degenerations of spherical subalgebras and spherical roots

In this paper, we obtain several structure results for a class of spherical subgroups of connected reductive complex algebraic groups that extends the class of strongly solvable spherical subgroups. Based on these results, we construct certain one-parameter degenerations of the Lie algebras corresponding to such subgroups. As an application, we exhibit explicit algorithms for computing the set of spherical roots of such a spherical subgroup.

A cluster structure on the coordinate ring of partial flag varieties

The main goal of this paper is to show that the (multi-homogeneous) coordinate ring of a partial flag variety C[G/PK−] contains a cluster algebra if G is any semisimple complex algebraic group. We use derivation properties and a special lifting map to prove that the cluster algebra structure A of the coordinate ring C[NK] of a Schubert cell constructed by Goodearl and Yakimov can be lifted, in an explicit way, to a cluster structure Aˆ living in the coordinate ring of the corresponding partial flag variety. Then we use a minimality condition to prove that the cluster algebra Aˆ is equal to C[G/PK−] after localizing some special minors.

HALF POTENTIAL ON GEOMETRIC CRYSTALS AND CONNECTEDNESS OF CELLULAR CRYSTALS

For any simply connected, simple complex algebraic group, we define upper/lower half-decorated geometric crystals and show that their tropicalization will be upper/lower normal Kashiwara's crystals. In particular, we show that the tropicalization of the half-decorated geometric crystal on the big Bruhat cell \( \left(=B{\overline{w}}_0:= {B}^{-}\cap U{\overline{w}}_0U\right) \) is isomorphic to the Langlands dual crystal B(∞) of the nilpotent-half subalgebra of quantum group. As an application, we shall show that any cellular crystal associated with a reduced word is connected in the sense of a crystal graph.

The Action of the Thompson Group $F$ on Infinite Trees

We construct an action of the Thompson group $F$ on a compact space built from pairs of infinite, binary rooted trees. The action arises as an $F$-equivariant compactification of the action of $F$ by translations on one of its homogeneous spaces, $F/H_2$, corresponding to a certain subgroup $H_2$ of $F$. The representation of $F$ on the Hilbert space $\ell^2(F/H_2)$ is faithful on the complex group algebra $\mathbb{C}[F]$.

Regular Functions on the 𝐾-nilpotent cone

Let G G be a complex reductive algebraic group with Lie algebra g \mathfrak {g} and let G R G_{\mathbb {R}} be a real form of G G with maximal compact subgroup K R K_{\mathbb {R}} . Associated to G R G_{\mathbb {R}} is a K × C × K \times \mathbb {C}^{\times } -invariant subvariety N θ \mathcal {N}_{\theta } of the (usual) nilpotent cone N ⊂ g ∗ \mathcal {N} \subset \mathfrak {g}^* . In this article, we will derive a formula for the ring of regular functions C [ N θ ] \mathbb {C}[\mathcal {N}_{\theta }] as a representation of K × C × K \times \mathbb {C}^{\times } . Some motivation comes from Hodge theory. In [Hodge theory and unitary representations of reductive Lie groups, Frontiers of Mathematical Sciences, Int. Press, Somerville, MA, 2011, pp. 397–420], Schmid and Vilonen use ideas from Saito’s theory of mixed Hodge modules to define canonical good filtrations on many Harish-Chandra modules (including all standard and irreducible Harish-Chandra modules). Using these filtrations, they formulate a conjectural description of the unitary dual. If G R G_{\mathbb {R}} is split, and X X is the spherical principal series representation of infinitesimal character 0 0 , then conjecturally g r ( X ) ≃ C [ N θ ] gr(X) \simeq \mathbb {C}[\mathcal {N}_{\theta }] as representations of K × C × K \times \mathbb {C}^{\times } . So a formula for C [ N θ ] \mathbb {C}[\mathcal {N}_{\theta }] is an essential ingredient for computing Hodge filtrations.

The structure of algebraic Baer ∗-algebras

The purpose of this note is to describe when a general complex algebraic $^*$-algebra is pre-$C^*$-normed, and to investigate their structure when the $^*$-algebras are Baer $^*$-rings in addition to algebraicity. As a main result we prove the following theorem for complex algebraic Baer $^*$-algebras: every $^*$-algebra of this kind can be decomposed as a direct sum $M\oplus B$, where $M$ is a finite dimensional Baer $^*$-algebra and $B$ is a commutative algebraic Baer $^*$-algebra. The summand $M$ is $^*$-isomorphic to a finite direct sum of full complex matrix algebras of size at least $2\times2$. The commutative summand $B$ is $^*$-isomorphic to the linear span of the characteristic functions of the clopen sets in a Stonean topological space. As an application we show that a group $G$ is finite exactly when the complex group algebra $\mathbb{C}[G]$ is an algebraic Baer $^*$-algebra.

Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres

Let $\Sigma$ be a finite type surface, and $G$ a complex algebraic simple Lie group with Lie algebra $\mathfrak{g}$. The quantum moduli algebra of $(\Sigma,G)$ is a quantization of the ring of functions of $X_G(\Sigma)$, the variety of $G$-characters of $\pi_1(\Sigma)$, introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid '90s. It can be realized as the invariant subalgebra of so-called graph algebras, which are $U_q(\mathfrak{g})$-module-algebras associated to graphs on $\Sigma$, where $U_q(\mathfrak{g})$ is the quantum group corresponding to $G$. We study the structure of the quantum moduli algebra in the case where $\Sigma$ is a sphere with $n+1$ open disks removed, $n\geq 1$, using the graph algebra of the "daisy" graph on $\Sigma$ to make computations easier. We provide new results that hold for arbitrary $G$ and generic $q$, and develop the theory in the case where $q=\epsilon$, a primitive root of unity of odd order, and $G={\rm SL}(2,{\mathbb C})$. In such a situation we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring $\mathcal{O}(G^n)$. We extend the quantum coadjoint action of De-Concini-Kac-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on $\mathcal{O}(G^n)$. We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ${\mathbb C}[X_G(\Sigma)]$ endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra $K_{\zeta}(\Sigma)$ at $\zeta:={\rm i}\epsilon^{1/2}$ with this quantum moduli algebra specialized at $q=\epsilon$.

A NOTE ON ÉTALE REPRESENTATIONS FROM NILPOTENT ORBITS

AbstractA linear étale representation of a complex algebraic group G is given by a complex algebraic G-module V such that G has a Zariski-open orbit in V and $\dim G=\dim V$ . A current line of research investigates which reductive algebraic groups admit such étale representations, with a focus on understanding common features of étale representations. One source of new examples arises from the classification theory of nilpotent orbits in semisimple Lie algebras. We survey what is known about reductive algebraic groups with étale representations and then discuss two classical constructions for nilpotent orbit classifications due to Vinberg and to Bala and Carter. We determine which reductive groups and étale representations arise in these constructions and we work out in detail the relation between these two constructions.

Coordinate rings and birational charts

Let G G be a semisimple simply connected complex algebraic group. Let U U be the unipotent radical of a Borel subgroup in G G . We describe the coordinate rings of U U (resp., G / U G/U , G G ) in terms of two (resp., four, eight) birational charts introduced by Lusztig [Total positivity in reductive groups, Birkhäuser Boston, Boston, MA, 1994; Bull. Inst. Math. Sin. (N.S.) 14 (2019), pp. 403–459] in connection with the study of total positivity.

$F$-theory over a Fano threefold built from $A_4$-roots

In a previous paper, the authors showed the advantages of building a $\mathbb{Z}_{2}$-action into an $F$-theory model $W_{4}/B_{3}$, namely the action of complex conjugation on the complex algebraic group with compact real form $E_{8}$. The goal of this paper is to construct the Fano threefold $B_{3}$ directly from the roots of $SU\left(5\right)$ in such a way that the action of complex conjugation is exactly the desired $\mathbb{Z}_{2}$-action and the quotient of this action on $W_{4}/B_{3}$ and its Heterotic dual have the phenomenologically correct invariants.

Poisson maps between character varieties: gluing and capping

Let G be a compact Lie group or a complex reductive affine algebraic group. We explore induced mappings between G-character varieties of surface groups by mappings between corresponding surfaces. It is shown that these mappings are generally Poisson. We also given an effective algorithm to compute the Poisson bi-vectors when G=SL(2,C). We demonstrate this algorithm by explicitly calculating the Poisson bi-vector for the 5-holed sphere, the first example for an Euler characteristic -3 surface.

On the multiplication of spherical functions of reductive spherical pairs of type A

AbstractLetGbe a simple complex algebraic group, and let$K \subset G$be a reductive subgroup such that the coordinate ring of$G/K$is a multiplicity-freeG-module. We consider theG-algebra structure of$\mathbb C[G/K]$and study the decomposition into irreducible summands of the product of irreducibleG-submodules in$\mathbb C[G/K]$. When the spherical roots of$G/K$generate a root system of type$\mathsf A$, we propose a conjectural decomposition rule, which relies on a conjecture of Stanley on the multiplication of Jack symmetric functions. With the exception of one case, we show that the rule holds true whenever the root system generated by the spherical roots of$G/K$is a direct sum of subsystems of rank 1.

Almost dominant generalized slices and convolution diagrams over them

Let G be a connected reductive complex algebraic group with a maximal torus T. We denote by Λ the coweight lattice of T. Let Λ+⊂Λ be the submonoid of dominant coweights. For λ∈Λ+,μ∈Λ,μ⩽λ, in [7], authors defined a generalized transversal slice W‾μλ. This is an algebraic variety of the dimension 〈2ρ∨,λ−μ〉, where 2ρ∨ is the sum of positive roots of G. In this paper, we construct an isomorphism W‾μλ≃W‾μ+λ×A〈2ρ∨,μ+−μ〉 for μ∈Λ such that 〈α∨,μ〉⩾−1 for any positive root α∨, here μ+∈Wμ is the dominant representative in the Weyl group orbit of μ. We consider the example when λ is minuscule, μ∈Wλ and describe natural coordinates, Poisson structure on W‾μλ≃A〈2ρ∨,λ−μ〉 and its T×C×-character. We apply these results to compute T×C×-characters of tangent spaces at fixed points of convolution diagrams W˜μλ_ with minuscule λi. We also apply our results to construct open coverings by affine spaces of convolution diagrams W˜μλ_ over slices with μ such that 〈α∨,μ〉⩾−1 for any positive root α∨ and minuscule λi and to compute Poincaré polynomials of such convolution diagrams W˜μλ_.