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Register a new userRamified Satake Isomorphisms for strongly parabolic characters
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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.
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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 $G$.
We give an explicit construction of test vectors for $T$-equivariant linear functionals on representations $Pi$ of $GL_2$ of a $p$-adic field $F$, where $T$ is a non-split torus. Of particular interest is the case when both the representations are ramified; we completely solve this problem for principal series and Steinberg representations of $GL_2$, as well as for depth zero supercuspidals over $mathbf{Q}_p$. A key ingredient is a theorem of Casselman and Silberger, which allows us to quickly reduce almost all cases to that of the principal series, which can be analyzed directly. Our method shows that the only genuinely difficult cases are the characters of $T$ which occur in the primitive part (or type) of $Pi$ when $Pi$ is supercuspidal. The method to handle the depth zero case is based on modular representation theory, motivated by considerations from Deligne-Lusztig theory and the de Rham cohomology of Deligne-Lusztig-Drinfeld curves. The proof also reveals some interesting features related to the Langlands correspondence in characteristic $p$. We show in particular that the test vector problem has an obstruction in characteristic $p$ beyond the root number criterion of Waldspurger and Tunnell, and exhibits an unexpected dichotomy related to the weights in Serres conjecture and the signs of standard Gauss sums.
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.
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 $mathscr{C}_mathfrak{g}$.
Lascoux stated that the type A Kostka-Foulkes polynomials K_{lambda,mu}(t) expand positively in terms of so-called atomic polynomials. For any semisimple Lie algebra, the former polynomial is a t-analogue of the multiplicity of the dominant weight mu in the irreducible representation of highest weight lambda. We formulate the atomic decomposition in arbitrary type, and view it as a strengthening of the monotonicity of K_{lambda,mu}(t). We also define a combinatorial version of the atomic decomposition, as a decomposition of a modified crystal graph. We prove that this stronger version holds in type A (which provides a new, conceptual approach to Lascouxs statement), in types B, C, and D in a stable range for t=1, as well as in some other cases, while we conjecture that it holds more generally. Another conjecture stemming from our work leads to an efficient computation of K_{lambda,mu}(t). We also give a geometric interpretation.
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