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Register a new userThe Jordan-Holder series of the locally analytic Steinberg representation
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We determine the composition factors of a Jordan-Holder series including multiplicities of the locally analytic Steinberg representation. For this purpose we prove the acyclicity of the evaluated locally analytic Tits complex giving rise to the Steinberg representation. Further we describe some analogue of the Jacquet functor applied to the irreducible principal series representation constructed by Orlik and Strauch.
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We prove that the Steinberg representation of a connected reductive group over an infinite field is irreducible. For finite fields, this is a classical theorem of Steinberg and Curtis.
In this paper we give a simple description of DT-invariants of double quivers without potential as the multiplicity of the Steinberg character in some representation associated with the quiver. When the dimension vector is indivisible we use this description to express these DT-invariants as the Poincare polynomial of some singular quiver varieties. Finally we explain the connections with previous work of Hausel-Letellier-Villegas where DT-invariants are expressed as the graded multiplicities of the trivial representation of some Weyl group in the cohomology of some non-singular quiver varieties attached to an extended quiver.
We present sum-sides for principal characters of all standard (i.e., integrable and highest-weight) irreducible modules for the affine Lie algebra $A_2^{(2)}$. We use modifications of five known Bailey pairs; three of these are sufficient to obtain all the necessary principal characters. We then use the technique of Bailey lattice appropriately extended to include out-of-bounds values of one of the parameters, namely, $i$. We demonstrate how the sum-sides break into six families depending on the level of the modules modulo 6, confirming a conjecture of McLaughlin--Sills.
We consider the unital associative algebra $mathcal{A}$ with two generators $mathcal{X}$, $mathcal{Z}$ obeying the defining relation $[mathcal{Z},mathcal{X}]=mathcal{Z}^2+Delta$. We construct irreducible tridiagonal representations of $mathcal{A}$. Depending on the value of the parameter $Delta$, these representations are associated to the Jacobi matrices of the para-Krawtchouk, continuous Hahn, Hahn or Jacobi polynomials.
We show that a Jordan-Holder theorem holds for appropriately defined composition series of finite dimensional Hopf algebras. This answers an open question of N. Andruskiewitsch. In the course of our proof we establish analogues of the Noether isomorphism theorems of group theory for arbitrary Hopf algebras under certain faithful (co)flatness assumptions. As an application, we prove an analogue of Zassenhaus butterfly lemma for finite dimensional Hopf algebras. We then use these results to show that a Jordan-Holder theorem holds as well for lower and upper composition series, even though the factors of such series may be not simple as Hopf algebras.
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