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Ana Prlić

Publications and source records attributed to Ana Prlić.

10 recordsLinked to original sources
arXiv2026

Unitary highest weight modules for $\mathfrak{su}(p, q)$ and $\mathfrak{so}^{*}(2n)$ with fixed integral infinitesimal character

We classify unitary highest weight modules with a given integral infinitesimal character for the real Lie algebras $\mathfrak{su}(p,q)$ and $\mathfrak{so}^*(2n)$. We treat both regular and singular cases. For $\mathfrak{su}(p,q)$ we identify the unitarizable modules in the Hasse diagrams of the highest weight orbit. Analogous results for the other Hermitian Lie algebras were given in our earlier publications.

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arXiv2025

On the classification of unitary highest weight modules in the exceptional cases

In our previous paper, we gave a complete classification of the unitary highest weight modules for the universal covers of the Lie groups $Sp(2n, \mathbb{R}), SO^{*}(2n)$ and $SU(p, q)$, using the Dirac inequality and the so called PRV product. In this paper, we complete the classification of the unitary highest weight modules for the remaining cases; i.e., universal covers of the Lie groups $SO_{e}(2, n)$, $E_{6(-14)}$ and $E_{7(-25)}$. We also describe unitary highest weight modules with given infinitesimal characters.

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arXiv2025

On the classification of unitary highest weight modules

In the 1980s, Enright, Howe and Wallach [EHW] and independently Jakobsen [J] gave a complete classification of the unitary highest weight modules. In this paper we give a more direct and elementary proof of the same result for the (universal covers of the) Lie groups $Sp(2n, \mathbb{R}), SO^{*}(2n)$ and $SU(p, q)$. We also show how to describe the set of unitary highest weight modules with a given infinitesimal character.

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arXiv2022

Dirac inequality for highest weight Harish-Chandra modules I

Let $G$ be a connected simply connected noncompact classical simple Lie group of Hermitian type. Then $G$ has unitary highest weight representations. The proof of the classification of unitary highest weight representations of $G$ given by Enright, Howe and Wallach is based on the Dirac inequality of Parthasarathy, Jantzen's formula and Howe's theory of dual pairs where one group in the pair is compact. In this paper we focus on the Dirac inequality which can be used to prove the classification in a more direct way.

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arXiv2019

Construction of discrete series representations of $SO_{e}(4, 1)$ via algebraic Dirac induction

The notion of algebraic Dirac induction was introduced by P. Pandžić and D. Renard. This is a construction which gives representations with prescribed Dirac cohomology. They proved that all holomorphic discrete series representations can be constructed via Dirac induction. All discrete series representations of the group $SO_{e}(4,1)$ are nonholomorphic. In this paper we prove that they can also be constructed using algebraic Dirac induction.

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arXiv2018

Classification of $A_{\mathfrak{q}}(λ)$ modules by their Dirac cohomology for type $D$, $G_2$ and $\mathfrak{sp}(2n,\mathbb{R})$

Let $G$ be a connected real reductive group with maximal compact subgroup $K$ of the same rank as $G$. In the recent paper of Huang, Pandžić and Vogan, it was shown that the admissible $Θ$--stable parabolic subalgebras $\mathfrak{q}$ of $\mathfrak{g}$ are in one-to-one correspodence with the faces of $W ρ$ intersecting the $\mathfrak{k}$--dominant Weyl chamber and that $A_{\mathfrak{q}}(0)$--modules can be classified by their Dirac cohomology in geometric terms. They described in detail the cases when $\mathfrak{g}_0$ is of type $A$, $B$, $F$ and $C$ except for $\mathfrak{g}_0 = \mathfrak{sp}(2n, \mathbb{R})$. We will describe faces corresponding to $A_{\mathfrak{q}}(0)$--modules for $\mathfrak{g}_0 = \mathfrak{sp}(2n, \mathbb{R})$ and for $\mathfrak{g}_0$ of type $D$ and $G_2$.

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arXiv2017

The centralizer of $K$ in $U(\mathfrak{g}) \otimes C(\mathfrak{p})$ for the group $SO_e(4,1)$

Let $G$ be the Lie group $SO_e(4,1)$, with maximal compact subgroup $K = S(O(4) \times O(1))_e\cong SO(4)$. Let $\mathfrak{g}=\mathfrak{so}(5,\mathbb{C})$ be the complexification of the Lie algebra $\mathfrak{g}_0 = \mathfrak{so}(4,1)$ of $G$, and let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$. Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be the Cartan decomposition of $\mathfrak{g}$, and $C(\mathfrak{p})$ the Clifford algebra of $\mathfrak{p}$ with respect to the trace form $B(X, Y) = \text{tr}(XY)$ on $\mathfrak{p}$. In this paper we give explicit generators of the algebra $(U(\mathfrak{g}) \otimes C(\mathfrak{p}))^{K}$.

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arXiv2015

Algebraic Dirac induction for nonholomorphic discrete series of SU(2, 1)

In a joint paper P. Pandžić and D. Renard proved that holomorphic and antiholomorphic discrete series representations can be constructed via algebraic Dirac induction. The group $SU(2,1)$, except for those two types, also has a third type of discrete series representations that are neither holomorphic nor antiholomorphic. In this paper we show that nonholomorphic discrete series representations of the group $SU(2,1)$ can also be constructed using algebraic Dirac induction.

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arXiv2014

K-invariants in the algebra U(g) $\otimes$ C(p) for the group SU(2,1)

Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be the Cartan decomposition of the complexified Lie algebra $\mathfrak{g}=\mathfrak{sl}(3,\mathbb{C})$ of the group $G=SU(2,1)$. Let $K=S(U(2) \times U(1))$; so $K$ is a maximal compact subgroup of $G$. Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$, and let $C(\mathfrak{p})$ be the Clifford algebra with respect to the trace form $B(X,Y)=\text{tr}(XY)$ on $\mathfrak{p}$. We are going to prove that the algebra of K-invariants in $U(\mathfrak{g}) \otimes C(\mathfrak{p})$ is generated by five explicitly given elements. This is useful for studying algebraic Dirac induction for $(\mathfrak{g},K)$-modules. Along the way we will also recover the (well known) structure of the algebra $U(\mathfrak{g})^K$.

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