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Let H be any reductive p-adic group. We introduce a notion of cuspidality for enhanced Langlands parameters for H, which conjecturally puts supercuspidal H-representations in bijection with such L-parameters. We also define a cuspidal…

Let K be a finite extension of Q_p with residue field F_q and let P(T) = T^d + a_{d-1}T^{d-1} + ... +a_1 T, where d is a power of q and a_i is in the maximal ideal of K for all i. Let u_0 be a uniformizer of O_K and let {u_n}_{n \geq 0} be…

In this paper we study non-abelian extensions of a Lie group $G$ modeled on a locally convex space by a Lie group $N$. The equivalence classes of such extension are grouped into those corresponding to a class of so-called smooth outer…

We study the Balmer spectrum of the category of finite G-spectra for a compact Lie group G, extending the work for finite G by Strickland, Balmer-Sanders, Barthel-Hausmann-Naumann-Nikolaus-Noel-Stapleton and others. We give a description of…

We introduce the notion of totally reflexive extension of rings. It unifies Gorenstein orders and Frobenius extensions. We prove that for a totally reflexive extension, a module over the extension ring is totally reflexive if and only if…

The main result asserts: Let $G$ be a reductive, affine algebraic group and let $(\rho ,V)$ be a regular representation of $G$. Let $X$ be an irreducible $\mathbb{C}^{ \times } G$ invariant Zariski closed subset such that $G$ has a closed…

This paper computes the obstruction to the existence of equivariant extensions of basic gerbes over non-simply connected compact simple Lie groups. By modifying a (finite dimensional) construction of Gaw\c{e}dzki-Reis [J. Geom. Phys.…

Basing ourselves on Janelidze and Kelly's general notion of central extension, we study universal central extensions in the context of semi-abelian categories. Thus we unify classical, recent and new results in one conceptual framework. The…

In this note, starting with any group homomorphism $f\colon\Gamma\to G$, which is surjective upon abelianization, we construct a universal central extension $u\colon U\twoheadrightarrow G,$ UNDER $\Gamma$ with the same surjective property,…

Suppose $R$ is a $\mathbb{Q}$-Gorenstein $F$-finite and $F$-pure ring of prime characteristic $p>0$. We show that if $I\subseteq R$ is a compatible ideal (with all $p^{-e}$-linear maps) then there exists a module finite extension $R\to S$…

In this paper we show that the Poisson analogue of the Noether's Problem has a positive solution for essentially all finite symplectic reflection groups - the analogue of complex reflection groups in the symplectic world. Our proofs are…

The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group $G$ on a reflexive…

Using symmetric algebras we simplify (and slightly strengthen) the Bruns-Eisenbud-Evans "generalized principal ideal theorem" on the height of order ideals of non-minimal generators in a module. We also obtain a simple proof and an…

Fix an abelian variety $A$ of dimension $g\geq 1$ defined over a number field $K$. For each prime $\ell$, the Galois action on the $\ell$-power torsion points of $A$ induces a representation $\rho_{A,\ell}\colon Gal_K \to…

We prove that every reflexive abelian group $G$ such that its dual group $G^\wedge$ has the $qc$-Glicksberg property is a Mackey group. We show that a reflexive abelian group of finite exponent is a Mackey group. We prove that, for a…

Let K be any compact set in the complex plane that has a connected complement, let A(K) be the uniforn algebra of all continuous complex functions on K that are holomorphic on the interior of K, let bK be the topological boundary of K, let…

By [R. Bautista, P. Gabriel, A.V Roiter., L. Salmeron, Representation-finite algebras and multiplicative basis. Invent. Math. 81 (1985) 217-285.], a finite-dimensional algebra having finitely many isoclasses of indecomposable…

A direct application of Zorn's Lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_p^n\to \mathbb{Q}_p^\ell$ has an extension to a Lipschitz map $\widetilde f: \mathbb{Q}_p^n\to \mathbb{Q}_p^\ell$. This is analogous, but more easy,…

We prove that the canonical dimension of a coadmissible representation of a semisimple $p$-adic Lie group in a $p$-adic Banach space is either zero or at least half the dimension of a non-zero coadjoint orbit. To do this we establish…

We generalize Dirichlet's $S$-unit theorem from the usual group of $S$-units of a number field $K$ to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over $S$. Specifically, we demonstrate…