On a determinantal formula of Tadić. (English) Zbl 1288.22013
Let \(F\) denote a non-Archimedean local field and \(\mathrm{GL}(n, F)\) the general linear group of rank \(n\) over \(F\). For an irreducible cuspidal representation \(\sigma\) of \(\mathrm{GL}(m,F)\) and a segment \([a, b]\), we denote by \(\Delta ([a,b])\) the unique irreducible quotient of the induced representation \(\nu^{a} \sigma \times \cdots \times \nu^{b}\). If \(\Delta_{i}\) is the representation attached to the segment \([a_{i}, b_{i}]\), \(i = 1, 2, \ldots, t\), and \(\Delta_{i}\) does not precede \(\Delta_{j}\) for \(i < j\), then the induced representation has a unique irreducible (Langlands) quotient, denoted by \(L(\Delta_{1}, \Delta_{2}, \ldots, \Delta_{t})\).
In the paper under review, the authors introduce the following concept of ladder representation:
\(\Delta_{1}, \Delta_{2}, \ldots, \Delta_{t}\) is a ladder if \(a_{1} > a_{2} > \cdots > a_{t}\) and \(b_{1} > b_{2} > \cdots > b_{t}\) and in that case \(L(\Delta_{1}, \Delta_{2}, \ldots, \Delta_{t})\) is called the ladder representation. These representations serve as basic building blocks for the unitary dual of \(\mathrm{GL}(n,F)\) and contain the Speh representation as a particulary important subclass. The authors extend many results about the Speh representation to the class of ladder representations, studying their structure and using the Jacquet module techniques. Furthermore, it appears that it is not enough to study only the semisimplification of the Jacquet module, but also the structure of its submodules.
Let us denote by \(S_{t}\) the symmetric group on \(\{ 1, 2, \ldots, t \}\). For \(w \in S_{t}\), let \(\mathcal{I}_{w}\) stand for \[ \Delta([a_{w(1)}, b_{1}]) \times \Delta([a_{w(2)}, b_{2}]) \times \cdots \times \Delta([a_{w(t)}, b_{t}]). \] Further, let \(s_{k}\) denote the transposition \((k, k+1)\). It is proved in the paper that, for a ladder \(\Delta_{1}, \Delta_{2}, \ldots, \Delta_{t}\), the maximal proper submodule of \(\Delta_{1} \times \Delta_{2} \times \cdots \times \Delta_{t}\) equals \(\sum_{k=1}^{t-1} \mathcal{I}_{s_{k}}\). Also, the ladder representation \(L(\Delta_{1}, \Delta_{2}, \ldots, \Delta_{t})\) satisfies Tadić’s determinantal formula \[ L(\Delta_{1}, \Delta_{2}, \ldots, \Delta_{t}) = \sum_{w \in S_{t}} \text{sgn}(w) \mathcal{I}_{w}, \] which has previously been known for the Speh representation. These results allow one also to compute the full (Bernstein-Zelevinsky) derivative of a ladder representation in terms of subordinate ladder represenations.
In the paper under review, the authors introduce the following concept of ladder representation:
\(\Delta_{1}, \Delta_{2}, \ldots, \Delta_{t}\) is a ladder if \(a_{1} > a_{2} > \cdots > a_{t}\) and \(b_{1} > b_{2} > \cdots > b_{t}\) and in that case \(L(\Delta_{1}, \Delta_{2}, \ldots, \Delta_{t})\) is called the ladder representation. These representations serve as basic building blocks for the unitary dual of \(\mathrm{GL}(n,F)\) and contain the Speh representation as a particulary important subclass. The authors extend many results about the Speh representation to the class of ladder representations, studying their structure and using the Jacquet module techniques. Furthermore, it appears that it is not enough to study only the semisimplification of the Jacquet module, but also the structure of its submodules.
Let us denote by \(S_{t}\) the symmetric group on \(\{ 1, 2, \ldots, t \}\). For \(w \in S_{t}\), let \(\mathcal{I}_{w}\) stand for \[ \Delta([a_{w(1)}, b_{1}]) \times \Delta([a_{w(2)}, b_{2}]) \times \cdots \times \Delta([a_{w(t)}, b_{t}]). \] Further, let \(s_{k}\) denote the transposition \((k, k+1)\). It is proved in the paper that, for a ladder \(\Delta_{1}, \Delta_{2}, \ldots, \Delta_{t}\), the maximal proper submodule of \(\Delta_{1} \times \Delta_{2} \times \cdots \times \Delta_{t}\) equals \(\sum_{k=1}^{t-1} \mathcal{I}_{s_{k}}\). Also, the ladder representation \(L(\Delta_{1}, \Delta_{2}, \ldots, \Delta_{t})\) satisfies Tadić’s determinantal formula \[ L(\Delta_{1}, \Delta_{2}, \ldots, \Delta_{t}) = \sum_{w \in S_{t}} \text{sgn}(w) \mathcal{I}_{w}, \] which has previously been known for the Speh representation. These results allow one also to compute the full (Bernstein-Zelevinsky) derivative of a ladder representation in terms of subordinate ladder represenations.
Reviewer: Ivan Matić (Osijek)
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 22E35 | Analysis on \(p\)-adic Lie groups |
