The minimal polynomials of unipotent elements in irreducible representations of the classical groups in odd characteristic

On the Jordan block structure of images of some unipotent elements in modular irreducible representations of the classical algebraic groups

For irreducible rational representations of the classical algebraic groups in characteristic p > 2, which are not equivalent to the composition of a group morphism and the standard representation, it is proved that usually the image of a unipotent element of order ps+1 > p has at least two Jordan blocks of size > ps; all exceptions are indicated explicitly. As a corollary, irreducible rational representations of these groups whose images contain unipotent elements with just one Jordan block of size > 1 are classified.

Let π be an irreducible smooth complex representation of a general linear p-adic group and let σ be an irreducible complex supercuspidal representation of a classical p-adic group of a given type, so that π ⨁ σ is a representation of a standard Levi subgroup of a p-adic classical group of higher rank. We show that the reducibility of the representation of the appropriate p-adic classical group obtained by (normalized) parabolic induction from π ⨁ σ does not depend on σ, if σ is “separated” from the supercuspidal support of π. (Here, “separated” means that, for each factor ρ of a representation in the supercuspidal support of π, the representation parabolically induced from ρ ⨁ σ is irreducible.) This was conjectured by E. Lapid and M. Tadic. (In addition, they proved, using results of C. Jantzen, that this induced representation is always reducible if the supercuspidal support is not separated.) More generally, we study, for a given set I of inertial orbits of supercuspidal representations of p-adic general linear groups, the category CI,σ of smooth complex finitely generated representations of classical p-adic groups of fixed type, but arbitrary rank, and supercuspidal support given by σ and I, and show that this category is equivalent to a category of finitely generated right modules over a direct sum of tensor products of extended affine Hecke algebras of type AB and D and establish functoriality properties, relating categories with disjoint I’s. In this way, we extend results of C. Jantzen who proved a bijection between irreducible representations corresponding to these categories. The proof of the above reducibility result is then based on Hecke algebra arguments, using Kato’s exotic geometry.

The minimal polynomials of the images of the unipotent elements of non-prime order in the irreducible representations of an algebraic group of type F4 in characteristics 3 and 7 are found. This completes the solution of the minimal polynomial problem for unipotent elements in the irreducible representations of such a group in an odd characteristic.

Introduction. The purpose of the present paper is to determine the decomposition of the Kronecker product of two irreducible representations of the real 2X2 unimodular group into a continuous direct sum of irreducible representations. The irreducible unitary representations of this group have been determined first by V. A. Bargmann [l](1), and those of the 2X2 complex unimodular group by I. M. Gel'fand and M. A. Nalmark [3]. In both cases the list of these representations contains two continuous series; first, the principal continuous series, the members of which can be described by a pair (m, p) of two variables, m with a discrete, p with a continuous range; and secondly, the representations of the exceptional interval, characterized by a single parameter, varying over a finite interval. In the real case in addition to these there exists a discrete series of representations characterized by integers. Concerning the representations of the exceptional interval it has been proved that they do not occur in the decomposition of the left regular representations of these groups into a continuous direct sum of irreducible representations. The problem of finding the irreducible parts for the Kronecker product of two of these representations by the Reduction Theory of von Neumann [9] was taken up first by G. W. Mackey, in the complex case, for two factors taken from the principal series [4; 5]. W. F. Stinespring applied the same method to the discussion of the analogous case for the real group(2). Recently, M. A. Nalmark attacked the same problem in the complex case, and gives a complete discussion of all possibilities [10](3). In Parts I, II, and III of the present work we give the decomposition of the product of any two irreducible unitary representations of the real 2X2 unimodular group. To sketch our method, we restrict ourselves, for the sake

This is a survey of several models (including new models) of irreducible complementary series representations and their limits, special representations, for the groups and . These groups, whose geometrical meaning is well known, exhaust the list of simple Lie groups for which the identity representation is not isolated in the space of irreducible unitary representations (that is, which do not have the Kazhdan property) and hence there exist irreducible unitary representations of these groups, so-called `special representations', for which the first cohomology of the group with coefficients in these representations is non-trivial. For technical reasons it is more convenient to consider the groups and , and most of this paper is devoted to the group .The main emphasis is on the so-called commutative models of special and complementary series representations: in these models, the maximal unipotent subgroup is represented by multipliers in the case of , and by the canonical model of the Heisenberg representations in the case of . Earlier, these models were studied only for the group . They are especially important for the realization of non-local representations of current groups, which will be considered elsewhere.Substantial use is made of the `denseness' of the irreducible representations under study for the group : their restrictions to the maximal parabolic subgroup are equivalent irreducible representations. Conversely, in order to extend an irreducible representation of to a representation of , it is necessary to determine only one involution. For the group , the situation is similar but slightly more complicated.

For p > 2, odd Jordan block sizes of the images of regular unipotent elements from subsystem subgroups of type A2 in irreducible p-restricted representations for groups of type Ar over the field of characteristic p, the weights of which are locally small with respect to p, are found. The weight is called locally small if the double sum of its two neighboring coefficients is less than p. This result is part of a more general program investigating the behavior of unipotent elements in representations of the classical algebraic groups. It can be used to solve recognition problems for representations or linear groups by the presence of certain elements.

The minimal polynomials of images of unipotent elements in irreducible rational representations of a special linear group over an algebraically closed field of characteristic p > 2 are found. In particular, we show that the degree of such polynomial is equal to the order of an element provided the highest weight of a representation is in some sense large enough with respect to p.

Modular representations of the supergroup Q(n), I

Modular irreducible representations of the symmetric group as linear codes

This third chapter presents a brief and somewhat sketchy introduction to the theory of unitary representations of reductive and semisimple Lie groups G. The basic fact for an irreducible unitary representation π of G on a Hilbert space ℋ, is that every irreducible representation к of a maximal compact subgroup K ⊂ G has multiplicity m(к, π| к) ≤ dim к. This yields up the infinitesimal character χπ: L(g)→ ℂ and the distribution character θπ: C c ∞ (G) → ℂ, and consequently the differential equations $$z({\theta _\pi }) = \,{\chi _\pi }(z){\theta _\pi }$$ for $$z \in L{\text{(g)}}$$ which are the starting point for serious harmonic analysis on G.

Let G be a noncompact connected real semisimple Lie group with finite center, and let K be a maximal compact subgroup of G. Let $\mathfrak {g}$ and $\mathfrak {k}$ denote the respective complexified Lie algebras. Then every irreducible representation $\pi$ of $\mathfrak {g}$ which is semisimple under $\mathfrak {k}$ and whose irreducible $\mathfrak {k}$-components integrate to finite-dimensional irreducible representations of K is shown to be equivalent to a subquotient of a representation of $\mathfrak {g}$ belonging to the infinitesimal nonunitary principal series. It follows that $\pi$ integrates to a continuous irreducible Hilbert space representation of G, and the best possible estimate for the multiplicity of any finite-dimensional irreducible representation of $\mathfrak {k}$ in $\pi$ is determined. These results generalize similar results of Harish-Chandra, R. Godement and J. Dixmier. The representations of $\mathfrak {g}$ in the infinitesimal nonunitary principal series, as well as certain more general representations of $\mathfrak {g}$ on which the center of the universal enveloping algebra of $\mathfrak {g}$ acts as scalars, are shown to have (finite) composition series. A general module-theoretic result is used to prove that the distribution character of an admissible Hilbert space representation of G determines the existence and equivalence class of an infinitesimal composition series for the representation, generalizing a theorem of N. Wallach. The composition series of Weylgroup-related members of the infinitesimal nonunitary principal series are shown to be equivalent. An expression is given for the infinitesimal spherical functions associated with the nonunitary principal series. In several instances, the proofs of the above results and related results yield simplifications as well as generalizations of certain results of Harish-Chandra.

The dimensions of the Jordan blocks in the images of regular unipotent elements from subsystem subgroups of type C2 in p-restricted irreducible representations of groups of type Cn in characteristic p ≥ 11 with locally small highest weights are found. These results can be applied for investigating the behavior of unipotent elements in modular representations of simple algebraic groups and recognizing representations and linear groups. The article consists of 3 parts. In the first one, preliminary lemmas that are necessary for proving the principal results, are contained and the case where all weights of the restriction of a representation considered to a subgroup of type A1 containing a relevant unipotent element are less than p, is investigated.

The group and the minimal polynomial of a graph

Let G be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field F of characteristic p ≥ 0 {p\geq 0} , and let u ∈ G {u\in G} be a nonidentity unipotent element. Let ϕ be a non-trivial irreducible representation of G. Then the Jordan normal form of ϕ ⁢ ( u ) {\phi(u)} contains at most one non-trivial block if and only if G is of type G 2 {G_{2}} , u is a regular unipotent element and dim ⁡ ϕ ≤ 7 {\dim\phi\leq 7} . Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent element is represented by a matrix whose Jordan form has a single non-trivial block were determined by I. D. Suprunenko [21].