arXiv subjects

Pampa Paul

Publications and source records attributed to Pampa Paul.

6 recordsLinked to original sources

arXiv2025

Special cycles in compact locally Hermitian symmetric spaces of type III associated with the Lie group $SO_0(2,m)$

Let $G = SO_0(2,m),$ the connected component of the Lie group $SO(2,m);\ K = SO(2) \times SO(m),$ a maximal compact subgroup of $G;$ and $θ$ be the associated Cartan involution of $G.$ Let $X = G/K,\ \frak{g}_0$ be the Lie algebra of $G$ and $\frak{g} = \frak{g}_0^\mathbb{C}.$ In this article, we have considered the special cycles associated with all possible involutions of $G$ commuting with $θ.$ We have determined the special cycles which give non-zero cohomology classes in $H^*(Γ\backslash X; \mathbb{C})$ for some $θ$-stable torsion-free arithmetic uniform lattice $Γ$ in $G,$ by a result of Millson and Raghunathan. For each cohomologically induced representation $A_\frak{q}$ with trivial infinitesimal character, we have determined the special cycles for which the non-zero cohomology class has no $A_\frak{q}$-component, via Matsushima's isomorphism.

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arXiv2025

Borel-de Siebenthal Positive Root Systems

Let $G$ be a connected simple Lie group with finite centre, $K$ be a maximal compact subgroup of $G,$ and rank$(G)=$ rank$(K).$ Let $\frak{g}_0=$Lie$(G), \frak{k}_0=$Lie$(K) \subset \frak{g}_0, \frak{t}_0$ be a maximal abelian subalgebra of $\frak{k}_0, \frak{g}=\frak{g}_0^\mathbb{C}, \frak{k}=\frak{k}_0^\mathbb{C},$ and $\frak{h}=\frak{t}_0^\mathbb{C}.$ The existence of a Borel-de Siebenthal positive root system of $Δ(\frak{g}, \frak{h})$ is proved by Borel and de Siebenthal. In this article, we have determined all Borel-de Siebenthal positive root systems of $Δ(\frak{g}, \frak{h}),$ assuming the existence. As an application, we have determined the number of unitary equivalence classes of all Borel-de Siebenthal discrete series representations of $G$ (if $G/K$ is not Hermitian symmetric) with a fixed infinitesimal character.

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arXiv2025

Irreducible unitary representations with non-zero relative Lie algebra cohomology of a Lie group of type f4(4)

In this article, we have determined the irreducible unitary representations with non-zero relative Lie algebra cohomology and Poincare polynomials of cohomologies of these representations for a connected Lie group G with Lie algebra f4(4). We have also determined a necessary and sufficient condition for these representations to be discrete series representations and identified the discrete series representations and Borel-de Siebenthal discrete series representations among the irreducible unitary representations of G with non-zero relative Lie algebra cohomology.

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arXiv2023

Irreducible unitary representations with non-zero relative Lie algebra cohomology of the Lie group $SO_0(2,m)$

By a theorem of D. Wigner, an irreducible unitary representation with non-zero $(\frak{g},K)$-cohomology has trivial infinitesimal character, and hence up to unitary equivalence, these are finite in number. We have determined the number of equivalence classes of these representations and the Poincaré polynomial of cohomologies of these representations for the Lie group $SO_0(2,m)$ for any positive integer $m.$ We have also determined, among these, which are discrete series representations and holomorphic discrete series representations.

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arXiv2019

Geometric cycles in compact Riemannian locally symmetric spaces of type IV and automorphic representations of complex simple Lie groups

Let G be a connected complex simple Lie group with maximal compact subgroup U. Let g be the Lie algebra of G, and X = G/U be the associated Riemannian globally symmetric space of type IV. We have constructed three types of arithmetic uniform lattices in G, say of type 1, type 2, and type 3 respectively. If g is not equal to b_n, n>0, then for each 0 4), c_n (n > 5), or f_4. To prove these, we have simplified Kac's description of finite order automorphisms of g with respect to a Chevalley basis of g. Also we have determined some orientation preserving group action on some subsymmetric spaces of X.

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arXiv2013

Borel-de Siebenthal discrete series and associated holomorphic discrete series

Let G_0 be a simply connected noncompact real simple Lie group with maximal compact subgroup K_0. Assume that rank(G_0) = rank(K_0) so that G_0 has discrete series representations. If G_0/K_0 is Hermitian symmetric, there exists a relatively simple discrete series of G_0, called holomorphic discrete series. Now assume that G_0/K_0 is not Hermitian symmetric. In this case, we can define Borel-de Siebenthal discrete series of G_0 analogous to holomorphic discrete series. We consider a certain circle subgroup of K_0 whose centralizer L_0 is such that K_0/L_0 is an irreducible compact Hermitian symmetric space. Let (K_0)* be the dual of K_0 with respect to L_0. Then (K_0)*/L_0 is an irreducible non-compact Hermitian symmetric space dual to K_0/L_0. To each Borel-de Siebenthal discrete series of G_0, we can associate a holomorphic discrete series of (K_0)*. In this article, we address occurrence of common L_0-types between these two discrete series under certain conditions.

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