[Paper Review] Intertwining operators between line bundles on Grassmannians

Dmitry Gourevitch, Siddhartha Sahi|arXiv (Cornell University)|Mar 5, 2013

Homotopy and Cohomology in Algebraic Topology14 references3 citations

TL;DR

This paper explicitly determines the space of intertwining operators between degenerate principal series representations induced from maximal parabolic subgroups of $GL(n,F)$, where $F$ is a local field of arbitrary characteristic. It proves that the dimension of this space is at most one, and characterizes all cases where it is exactly one—classified as standard, mixed, or exceptional intertwiners—thereby establishing a complete multiplicity-free classification of such operators.

ABSTRACT

Let G=GL(n,F) where F is a local field of arbitrary characteristic, and let $π_1,π_2$ be representations induced from characters of two maximal parabolic subgroups $P_1,P_2$. We explicitly determine the space $Hom_G(π_1,π_2)$ of intertwining operators and prove that it has dimension at most 1 in all cases.

Motivation & Objective

• To determine the space $\mathrm{Hom}_{G_n}(\pi_1, \pi_2)$ of intertwining operators between degenerate principal series representations $\pi_1 = \chi_1 \times \chi_2$ and $\pi_2 = \chi_3 \times \chi_4$ on $GL(n,F)$.
• To classify all such intertwining operators explicitly, particularly identifying when they exist and their structure.
• To prove that the space of intertwining operators has dimension at most one in all cases, establishing a multiplicity-one property.
• To show that all such intertwiners arise from known geometric transforms (e.g., Radon, cosine transforms) or their generalizations, with no others existing.
• To classify the representations $\mathfrak{X} = (\chi_1, \chi_2, \chi_3, \chi_4)$ for which $\dim \mathrm{Hom}_{G_n}(\chi_1 \times \chi_2, \chi_3 \times \chi_4) = 1$, identifying standard, mixed, and exceptional cases.

Proposed method

• Use the framework of degenerate principal series representations induced from maximal parabolic subgroups $P_{p_1,p_2} \subset GL(n,F)$, with characters $\chi_1 \otimes \chi_2$ on $P$.
• Define the induced representation $\pi = \chi_1 \times \chi_2$ as smooth sections of a line bundle on the Grassmannian $G/P$, which parametrizes $p_1$-dimensional subspaces of $F^n$.
• Apply the theory of derivatives of representations and use the notion of a unique irreducible quotient/submodule to analyze the existence of non-zero intertwining maps.
• Use central twisting invariance: $H(\mathfrak{X}) \cong H(\psi\mathfrak{X})$ for any character $\psi$ of $F^\times$, to reduce the classification to canonical forms.
• Leverage the central character condition: $\psi_1(z)^{p_1}\psi_2(z)^{p_2} = \psi_3(z)^{p_3}\psi_4(z)^{p_4}$ for all $z \in F^\times$, to constrain possible representations.
• Use induction on $n = p_1 + p_2 = p_3 + p_4$, reducing the problem to smaller $n$ via the $H_0(\mathfrak{X})$ condition, and analyze cases based on whether $p_i = 1$.

Experimental results

Research questions

• RQ1 When does a non-zero intertwining operator exist between two degenerate principal series representations $\chi_1 \times \chi_2$ and $\chi_3 \times \chi_4$ on $GL(n,F)$?
• RQ2 What is the maximum possible dimension of the space $\mathrm{Hom}_{G_n}(\chi_1 \times \chi_2, \chi_3 \times \chi_4)$, and when is it exactly one?
• RQ3 Which representations $\mathfrak{X} = (\chi_1, \chi_2, \chi_3, \chi_4)$ yield non-trivial intertwiners, and can they be fully classified?
• RQ4 Are all known geometric intertwiners—such as Radon and cosine transforms—accounted for in this classification, or are there new ones?
• RQ5 How do central twists affect the existence and structure of intertwining operators, and can they be used to reduce the classification to canonical forms?

Key findings

• The space $\mathrm{Hom}_{G_n}(\pi_1, \pi_2)$ of intertwining operators between $\pi_1 = \chi_1 \times \chi_2$ and $\pi_2 = \chi_3 \times \chi_4$ has dimension at most one for all choices of $\chi_1, \chi_2, \chi_3, \chi_4$.
• The dimension is exactly one if and only if the representation quadruple $\mathfrak{X} = (\chi_1, \chi_2, \chi_3, \chi_4)$ is standard, mixed, or exceptional as defined in the paper.
• Standard intertwiners include identity maps ($\pi \dashrightarrow \pi$) and transposition maps ($\pi \dashrightarrow \tilde{\pi}$), which are always present.
• Mixed intertwiners arise from configurations like $\widetilde{\alpha_j} \dashrightarrow \alpha_i$ for $0 \leq i \neq j < k$, where $\alpha_i = [0,i) \times [i,k)$.
• Exceptional intertwiners exist for $F = \mathbb{R}$ or $\mathbb{C}$, including cases such as $1 \times \delta^i\varsigma \dashrightarrow \delta^i \times \varsigma$ for $GL_k(\mathbb{R})$ and similar for $GL_k(\mathbb{C})$.
• No other intertwiners exist beyond those in the standard, mixed, or exceptional classes—thus the classification is complete and exhaustive.