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Register a new userA note on Integral Satake isomorphisms
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Xinwen Zhu
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We formulate a Satake isomorphism for the integral spherical Hecke algebra of an unramified $p$-adic group $G$ and generalize the formulation to give a description of the Hecke algebra $H_G(V)$ of weight $V$, where $V$ is a lattice in an irreducible algebraic representation of $G$.
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For certain characters of the compact torus of a reductive $p$-adic group, which we call strongly parabolic characters, we prove Satake-type isomorphisms. Our results generalize those of Satake, Howe, Bushnell and Kutzko, and Roche.
This article constructs the Satake parameter for any irreducible smooth $J$-spherical representation of a $p$-adic group, where $J$ is any parahoric subgroup. This parametrizes such representations when $J$ is a special maximal parahoric subgroup. The main novelty is for groups which are not quasi-split, and the construction should play a role in formulating a geometric Satake isomorphism for such groups over local function fields.
We give a short proof based on Lusztigs generalized Springer correspondence of some results of [BrCh,BaCr,P].
Let k be an algebraically closed field. It is known that any stable equivalence between standard representation-finite self-injective k-algebras (without blocks of Lowey length 2) lifts to a standard derived equivalence, in particular, it is of Morita type. In this note, we show that the same holds for any stable equivalence between nonstandard representation-finite self-injective k-algebras. This settles an open question raised by H. Asashiba about twenty years ago.
Let ($mathfrak{g},mathsf{g})$ be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with $mathsf{g}$ being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories $mathscr{C}_{mathfrak{g}}$ and $mathscr{C}_{mathsf{g}}$ of finite-dimensional representations over the quantum loop algebras of $mathfrak{g}$ and $mathsf{g}$ respectively. As a consequence, we solve long-standing problems : the positivity of the analogs of Kazhdan-Lusztig polynomials and the positivity of the structure constants of the quantum Grothendieck rings for any non-simply-laced $mathfrak{g}$. In addition, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur-Weyl dualities, we prove the analog of Kazhdan-Lusztig conjecture (formulated in [H., Adv. Math., 2004]) for simple modules in remarkable monoidal subcategories of $mathscr{C}_{mathfrak{g}}$ for any non-simply-laced $mathfrak{g}$, and for any simple finite-dimensional modules in $mathscr{C}_{mathfrak{g}}$ for $mathfrak{g}$ of type $mathrm{B}_n$. In the course of the proof we obtain and combine several new ingredients. In particular we establish a quantum analog of $T$-systems, and also we generalize the isomorphisms of [H.-Leclerc, J. Reine Angew. Math., 2015] and [H.-O., Adv. Math., 2019] to all $mathfrak{g}$ in a unified way, that is isomorphisms between subalgebras of the quantum group of $mathsf{g}$ and subalgebras of the quantum Grothendieck ring of $mathscr{C}_mathfrak{g}$.
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