The multidimensional quantization procedure, proposed by the first author and its modifications (reduction to radicals and lifting on U(1)-coverings) give us a almost universal theoretical tools to find irreducible representations of Lie groups. By using this method and the root theory, we realize in this paper the representations of the degenerate principal series of symplectic groups.
In this paper, we study the reducibility of degenerate principal series of the simple, simply-connected exceptional group of type $E_6$. Furthermore, we calculate the maximal semi-simple subrepresentation and quotient of these representations.
In this paper, we study the reducibility of degenerate principal series of the simple, simply-connected exceptional group of type $E_8$. Furthermore, we calculate the maximal semi-simple subrepresentation and quotient of these representations for almost all cases.
In this paper, we study the degenerate principal series of a split, simply-connected, simple p-adic group of type $E_7$. We determine the points of reducibility and the maximal semi-simple subrepresentation at each point.
We define a set of PBW-semistandard tableaux that is in a weight preserving bijection with the set of monomials corresponding to integral points in the Feigin-Fourier-Littelmann-Vinberg polytope for highest weight modules of the symplectic Lie algebra. We then show that these tableaux parametrize bases of the homogeneous coordinate rings of the complete symplectic original and PBW degenerate flag varieties. From this construction, we provide explicit degenerate relations that generate the defining ideal of the PBW degenerate variety. These relations consist of type A degenerate Plücker relatio…
Let $G_n$ denote either the group $SO(2n+1, F)$, $Sp(2n, F)$, or $GSpin(2n+1, F)$ over a non-archimedean local field of characteristic different than two. We determine all composition factors of degenerate principal series of $G_n$, using methods based on the Aubert involution and known results on irreducible subquotiens of the generalized principal series of particular type.
We study $K$-types of degenerate principal series of ${\rm Sp}(n,\mathbb{C})$ by using two realisations of these infinite-dimensional representations. The first model we use is the classical compact picture; the second model is conjugate to the non-compact picture via an appropriate partial Fourier transform. In the first case we find a family of $K$-finite vectors that can be expressed as solutions of specific hypergeometric differential equations; the second case leads to a family of $K$-finite vectors whose expressions involve Bessel functions.
The goal of this paper is to study the link between the topology of the degenerate flag varieties and combinatorics of the Dellac configurations. We define three new classes of algebraic varieties closely related to the degenerate flag varieties of types A and C. The construction is based on the quiver Grassmannian realization of the degenerate flag varieties and odd symplectic and odd and even orthogonal groups. We study basic properties of our varieties; in particular, we construct cellular decomposition in all the three cases above (as well as in the case of even symplectic group). We also …
The algebra $\mathcal H:= H_{1,ν}(I_2(2m+1))$ of observables of the Calogero model based on the root system $I_2(2m+1)$ has an $m$-dimensional space of traces and an $(m+1)$-dimensional space of supertraces. In the preceding paper we found all values of the parameter $ν$ for which either the space of traces contains a~degenerate nonzero trace $tr_ν$ or the space of supertraces contains a~degenerate nonzero supertrace $str_ν$ and, as a~consequence, the algebra $\mathcal H$ has two-sided ideals: one consisting of all vectors in the kernel of the form $B_{tr_ν}(x,y)=tr_ν(xy)$ or another consistin…
We construct explicit local systems on the affine line in characteristic $p>2$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for all $n \ge 2$, and others whose geometric monodromy groups are the special unitary groups $SU_n(q)$ for all odd $n \ge 3$, and $q$ any power of $p$, in their total Weil representations. One principal merit of these local systems is that their associated trace functions are one-parameter families of exponential sums of a very simple, i.e., easy to remember, form. We also exhibit hypergeometric sheaves on $G_m$, whose geometric monodro…
