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Register a new userA quantum probabilistic approach to Hecke algebras for $mathfrak{p}$-adic ${rm PGL}_2$
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The subject of the present paper is an application of quantum probability to $p$-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for ${rm PGL}_2(F)$, where $F$ is a $p$-adic field. As a byproduct, we obtain a new proof of the Fourier inversion formula for ${rm PGL}_2(F)$.
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The higher rank Racah algebra $R(n)$ introduced recently is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra $sR(n)$, is then introduced. Using results from classical invariant theory, this $sR(n)$ algebra is shown to be isomorphic to the centralizer $Z_{n}(mathfrak{sl}_2)$ of the diagonal embedding of $U(mathfrak{sl}_2)$ in $U(mathfrak{sl}_2)^{otimes n}$. This leads to a first and novel presentation of the centralizer $Z_{n}(mathfrak{sl}_2)$ in terms of generators and defining relations. An explicit formula of its Hilbert-Poincare series is also obtained and studied. The extension of the results to the study of the special Askey-Wilson algebra and its higher rank generalizations is discussed.
The orthosymplectic super Lie algebra $mathfrak{osp}(1|,2ell)$ is the closest analog of standard Lie algebras in the world of super Lie algebras. We demonstrate that the corresponding $mathfrak{osp}(1|,2ell)$-Toda chain turns out to be an instance of a $BC_ell$-Toda chain. The underlying reason for this relation is discussed.
The affine evaluation map is a surjective homomorphism from the quantum toroidal ${mathfrak {gl}}_n$ algebra ${mathcal E}_n(q_1,q_2,q_3)$ to the quantum affine algebra $U_qwidehat{mathfrak {gl}}_n$ at level $kappa$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^n=kappa^2$. We discuss ${mathcal E}_n(q_1,q_2,q_3)$ evaluation modules. We give highest weights of evaluation highest weight modules. We also obtain the decomposition of the evaluation Wakimoto module with respect to a Gelfand-Zeitlin type subalgebra of a completion of ${mathcal E}_n(q_1,q_2,q_3)$, which describes a deformation of the coset theory $widehat{mathfrak {gl}}_n/widehat{mathfrak {gl}}_{n-1}$.
Let G be a split semi-simple p-adic group and let H be its Iwahori-Hecke algebra with coefficients in the algebraic closure k of the finite field with p elements. Let F be the affine flag variety over k associated with G. We show, in the simply connected simple case, that a torus-equivariant K-theory of F (with coefficients in k) admits an H-action by Demazure operators and that this provides a model for the regular representation of H.
We provide the localization procedure for monoidal categories by a real commuting family of braiders. For an element $w$ of the Weyl group, $mathscr{C}_w$ is a subcategory of modules over quiver Hecke algebra which categorifies the quantum unipotent coordinate algebra $A_q[mathfrak{n}(w)]$. We construct the localization $widetilde{mathscr{C}_w}$ of $mathscr{C}_w$ by adding the inverses of simple modules which correspond to the frozen variables in the quantum cluster algebra $A_q[mathfrak{n}(w)]$. The localization $widetilde{mathscr{C}_w}$ is left rigid and we expect that it is rigid.
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