scholarly journals On representations of reductive p-adic groups over ℚ-algebras

2020 ◽  
Vol 55 (2) ◽  
pp. 203-235
Author(s):  
Goran Muić ◽  
Keyword(s):  
Irreducible Representations ◽  
Complex Case ◽  
Complex Numbers ◽  
Smooth Complex

In this paper we study certain category of smooth modules for reductive p-adic groups analogous to the usual smooth complex representations but with the field of complex numbers replaced by a ℚ-algebra. We prove some fundamental results in these settings, and as an example we give a classification of admissible unramified irreducible representations using the reduction to the complex case.

scholarly journals REFINING THE CLASSIFICATION OF THE IRREPS OF THE 1D N-EXTENDED SUPERSYMMETRY

2008 ◽  
Vol 23 (01) ◽  
pp. 37-51 ◽  
Author(s):  
ZHANNA KUZNETSOVA ◽  
FRANCESCO TOPPAN
Keyword(s):  
Quantum Mechanics ◽  
Extended Supersymmetry ◽  
Supersymmetric Quantum Mechanics ◽  
Irreducible Representations

The linear finite irreducible representations of the algebra of the 1D N-Extended Supersymmetric Quantum Mechanics are discussed in terms of their "connectivity" (a symbol encoding information on the graphs associated to the irreps). The classification of the irreducible representations with the same fields content and different connectivity is presented up to N ≤ 8.

scholarly journals Local Langlands correspondence and ramification for Carayol representations

2019 ◽  
Vol 155 (10) ◽  
pp. 1959-2038
Author(s):  
Colin J. Bushnell ◽  
Guy Henniart
Keyword(s):  
Langlands Correspondence ◽  
Smooth Complex ◽  
Compact Field ◽  
Weil Group ◽  
Detailed Interpretation ◽  
Simple Character ◽  
Local Langlands Correspondence ◽  
Class Of Functions ◽  
Residual Characteristic

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with Weil group ${\mathcal{W}}_{F}$. Let $\unicode[STIX]{x1D70E}$ be an irreducible smooth complex representation of ${\mathcal{W}}_{F}$, realized as the Langlands parameter of an irreducible cuspidal representation $\unicode[STIX]{x1D70B}$ of a general linear group over $F$. In an earlier paper we showed that the ramification structure of $\unicode[STIX]{x1D70E}$ is determined by the fine structure of the endo-class $\unicode[STIX]{x1D6E9}$ of the simple character contained in $\unicode[STIX]{x1D70B}$, in the sense of Bushnell and Kutzko. The connection is made via the Herbrand function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ of $\unicode[STIX]{x1D6E9}$. In this paper we concentrate on the fundamental Carayol case in which $\unicode[STIX]{x1D70E}$ is totally wildly ramified with Swan exponent not divisible by $p$. We show that, for such $\unicode[STIX]{x1D70E}$, the associated Herbrand function satisfies a certain functional equation, and that this property essentially characterizes this class of representations. We calculate $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ explicitly, in terms of a classical Herbrand function arising naturally from the classification of simple characters. We describe exactly the class of functions arising as Herbrand functions $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6EF}}$, as $\unicode[STIX]{x1D6EF}$ varies over the set of totally wild endo-classes of Carayol type. In a separate argument, we derive a complete description of the restriction of $\unicode[STIX]{x1D70E}$ to any ramification subgroup and hence a detailed interpretation of the Herbrand function. This gives concrete information concerning the Langlands correspondence.

On Springer's correspondence for simple groups of type En (n = 6, 7, 8)

1982 ◽  
Vol 92 (1) ◽  
pp. 65-72 ◽  
Author(s):  
Dean Alvis ◽  
George Lusztig
Keyword(s):  
Irreducible Representation ◽  
Tensor Product ◽  
Algebraic Group ◽  
Irreducible Representations ◽  
Simple Groups ◽  
Complex Numbers ◽  
Unipotent Element ◽  
Reductive Algebraic Group ◽  
Dimension 2 ◽  
Injective Map

Let G be a connected reductive algebraic group over complex numbers. To each unipotent element u ε G (up to conjugacy) and to the unit representation of the group of components of the centralizer of u, Springer (11), (12) associates an irreducible representation of the Weyl group W of G. The tensor product of that representation with the sign representation will be denoted ρu. (This agrees with the notation of (5).) This representation may be realized as a subspace of the cohomology in dimension 2β(u) of the variety of Borel subgroups containing u, where β(u) = dim . For example, when u = 1, ρu is the sign representation of W. The map u → ρu defines an injective map from the set of unipotent conjugacy classes in G to the set of irreducible representations of W (up to isomorphism). Our purpose is to describe this map in the case where G is simple of type Eu (n = 6, 7, 8). (When G is classical or of type F4, this map is described by Shoji (9), (10); the case where G is of type G2 is contained in (11).

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