𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Weil Representations of the Symplectic Group

✍ Scribed by Fernando Szechtman

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
253 KB
Volume
208
Category
Article
ISSN
0021-8693

No coin nor oath required. For personal study only.

✦ Synopsis

We have deliberately favoured constructive proofs to existence arguments. In regards to linear representations, emphasis has been placed on matrices and linear transformations rather than modules and characters. As a result, every proposition asserting the existence of a certain object can be used as a recipe to construct that object.

1. CONSTRUCTION OF THE WEIL REPRESENTATIONS Definition of the Weil Representations

Weil representations arise from the interplay between Sp and the Heisenberg group H, upon which Sp acts as a group of automorphisms in such a way that all irreducible representations T of H of degree ) 1 are Sp-invariant. A Weil representation of Sp is one that intertwines the Sp-conjugates of a given T.

In this section we present a method, independent of Gerardin's, to Δ‡onstruct the Weil representations. We first recall the definition of the Ε½ Β² :. Heisenberg group H associated to the symplectic space V, , , that is,

with multiplication Β² : c , w c , w s c q c q w , w , w q w .

Ε½

.Ε½

. Ε½ .

we deduce that Z H s HЈ s c, 0 c g K . This gives q linear char-Ε½ . acters H Βͺ F*, where F s ‫ޑ‬ and is the primitive pth root of unity p p

Ε½a good deal of what we shall do can also be done when ‫ޑ‬ is replaced by . any field of characteristic / 2, p .

To obtain the remaining irreducible representations, we fix, once and for all, two totally isotropic subspaces, M and N, of V of maximal

πŸ“œ SIMILAR VOLUMES

Some Characterizations of the Weil Repre
Some Characterizations of the Weil Representations of the Symplectic and Unitary Groups
✍ Pham Huu Tiep; Alexander E. Zalesskii πŸ“‚ Article πŸ“… 1997 πŸ› Elsevier Science 🌐 English βš– 425 KB

Let G be a finite symplectic or unitary group. We characterize the Weil representations of G via their restriction to a standard subgroup. Then we complete the determination of complex representations of G with specific minimal polynomials of certain elements by showing that they coincide with the W

Symplectic Representations of Inertia Gr
Symplectic Representations of Inertia Groups
✍ A Silverberg; Yu.G Zarhin πŸ“‚ Article πŸ“… 2001 πŸ› Elsevier Science 🌐 English βš– 95 KB

2 g l Ε½ . Ε½ . which are not conjugate, even in GL Q , to a subgroup of Sp Z . 2 g l 2 g l Ε½ . However, it turns out and this is the main result of this paper that every 1 Silverberg thanks NSA and the Alexander-von-Humboldt Stiftung for financial support, and H. Lange and the Mathematics Institute

Antiholomorphic Representations for Orth
Antiholomorphic Representations for Orthogonal and Symplectic Quantum Groups
✍ Pavel Ε Ε₯ovΔ±́ček πŸ“‚ Article πŸ“… 1996 πŸ› Elsevier Science 🌐 English βš– 305 KB

The coadjoint orbits for the series B , C , and D are considered in the case when the base point is a multiple of a fundamental weight. A quantization of the big cell is suggested by means of introducing a )-algebra generated by holomorphic coordinate functions. Starting from this algebraic structu

Explicit Constructions of the Fundamenta
Explicit Constructions of the Fundamental Representations of the Symplectic Lie Algebras
✍ Robert G. Donnelly πŸ“‚ Article πŸ“… 2000 πŸ› Elsevier Science 🌐 English βš– 237 KB

We give two constructions for each fundamental representation of sp 2 n, ‫ރ‬ . We also present quantum versions of these constructions. These are explicit in the sense of the Gelfand᎐Tsetlin constructions of the irreducible representations of Ε½ . Ε½ . gl n, ‫ރ‬ : we explicitly specify the matrix elem