Irreducible representations of the classical algebraic groups with - - PowerPoint PPT Presentation

Irreducible representations

• f the classical algebraic groups

with p-large highest weights: properties of unipotent elements and restrictions to subgroups

• I. D. Suprunenko, Institute of Mathematics,

National Academy of Sciences of Belarus, Minsk, Belarus suprunenko@im.bas-net.by The main goal: to investigate special properties of irreducible representations of the clas- sical algebraic groups in positive characteristic with large highest weights with respect to the characteristic. Two groups of such properties: 1) those concerned with the behaviour of unipotent elements in representations, 2) the restrictions of representations under consideration to subsystem subgroups.

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p-large representations. For simple algebraic groups in characteristic p > 0 the notion of a p-large representation was introduced in 1997 (I.D. Suprunenko, On Jordan blocks of elements of order p in irreducible representations of classical groups with p-large highest weights, J. Algebra, 191(1997), 589–627) in order to distinguish some regularities that are specific for modular representations, do not depend upon a fixed characteristic, and hold when the highest weight is large enough with respect to the characteristic. This notion was introduced in the connection with the study of the behaviour of unipotent elements in irreducible representations. Every dominant weight µ can be written in the form µ = t

j=0 pjλj with p-restricted λj.

Set µ = t

j=0 λj (this weight is uniquely determined). We call an irreducible representation of

a simple algebraic group in characteristic p > 0 with highest weight ω p-large if the value of the weight ω on the maximal root of the group is at least p. If ω is p-restricted, then ω = ω. Let K be an algebraically closed field of characteristic p and G be a simply connected algebraic group over K. Denote by Fr the morphism of G associated with raising the elements of K to the pth power. For every positive integer j and rational irreducible representation ϕ of G the images of the representations ϕ and ϕj = ϕ · Frj coincide, the canonical Jordan form of the images of any unipotent element in ϕ and ϕj is the same. So there is no sense to call ϕj p-large if ϕ is not p-large. If ω is the highest weight of ϕ, then pjω is the highest weight of ϕj. That is why we pass to ω.

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• I. Properties of unipotent elements

in p-large representations. Lower estimates for the number of Jordan blocks of size p in the images of elements of order p in such representations in terms of the highest weight coefficients and the group rank are

• btained. This allows one to get estimates for coranks of the images of arbitrary unipotent

elements in relevant representations. Throughout the text we assume that G is a group of a classical type over K, r is the rank

• f G, ωi and αi, 1 ≤ i ≤ r, are its fundamental weights and simple roots, and ω(ϕ) is the

highest weight of an irreducible representation ϕ. For an irreducible representation ϕ denote by l(ϕ) the value of ω(ϕ) on the maximal root. We assume that p = 2 for G = Ar(K). If ω(ϕ) = r

i=1 aiωi, it is well known that

l(ϕ) =        r

i=1 ai

for G = Ar(K) or Cr(K), a1 + 2(a2 + . . . + ar−1) + ar for G = Br(K), a1 + 2(a2 + . . . + ar−2) + ar−1 + ar for G = Dr(K). In Theorems 1–3 and Corollary 4 ϕ is a p-restricted irreducible representation of G with highest weight ω = r

i=1 aiωi.

For G = Ar(K) set ω∗ = r

i=1 ar+1−iωi (this is the highest weight of the representation dual

to ϕ).

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Theorem 1 Let G = Ar(K), r > 8, and l(ϕ) ≥ p. 1). Assume that l(ϕ) ≥ p + 2, or r−2

i=3 ai = 0, or a2 + ar−1 > 1, or a2 + ar−1 = 0 and

l(ϕ) > p. Then for a unipotent element x ∈ G of order p the image ϕ(x) has at least (r − 2)3/8 Jordan blocks of size p. 2). Set Ω = {(p − 1)ω1 + ω2 + ωr, (p − 2)ω1 + 2ω2, a1ω1 + arωr, a1 + ar = p + 2}. If ω satisfies the assumptions of Item 1, ω and ω∗ ∈ Ω, and, furthermore, both ω and ω∗ = 2ω1 + ω3 for p = 3, then ϕ(x) has at least (l(ϕ) − p + 2)(r − 2)3/8 such blocks. Theorem 2 Let p > 2, G = Cr(K), r > 12, and l(ϕ) ≥ p. Assume that ω = (p−1)ω1+ω2. Then for a unipotent element x ∈ G of order p the image ϕ(x) has at least (r −1)3 Jordan blocks of size p. If ω = (p − 2)ω1 + 2ω2 and for p = 3 the weight ω = 2ω1 + ω3 as well, then ϕ(x) has at least (l(ϕ) − p + 2)(r − 1)3 such blocks. Theorem 3 Let p > 2, G = Br(K) or Dr(K), r ≥ 12 for G = Br(K) and r ≥ 14 for G = Dr(K). Assume that l(ϕ) ≥ p.

• 1. Suppose that r

i=4 ai = 0, or a3 > 1, or l(ϕ) > p and a2a3 = 0, or l(ϕ) > p + 1

and a2 > 2. Then for a unipotent element x ∈ G of order p the image ϕ(x) has at least 2(r − 2)3 Jordan blocks of size p.

• 2. Set

Ω = {(p − 1)ω1 + ω4, (p−1

2 )ω2 + ω3, (p − 2)ω1 + ω2 + ω3, ω1 + (p+1 2 )ω2; a1ω1 + 3ω2, a1 ≥ p − 3}.

If ω satisfies the assumptions of Item 1, ω ∈ Ω and, furthermore, ω = ω1 + ω4 for p = 3, then ϕ(x) has at least 2(l(ϕ) − p + 2)(r − 2)3 such blocks.

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Theorems 1–3 yield lower estimates for the coranks of the images of arbitrary unipotent elements in relevant representations. Set N(G) = (r − 2)3/8 for G = Ar(K), 2(r − 2)3 for G = Br(K) or Dr(K), and (r − 1)3 for G = Cr(K). For all types put N(G, ϕ) = (l(ϕ) − p + 2)N(G). Corollary 4 Let a representation ϕ satisfy the assumptions of Theorem 1, Theorem 2,

• r Theorem 3 for G = Ar(K), Cr(K), or Br(K) and Dr(K), respectively. Assume that

M is a G-module affording ϕ. Then for a nontrivial unipotent element x ∈ G one has dim(x − 1)M ≥ (p − 1)N(G). Moreover, if the assumptions of Item 2 of Theorem 1 or Theorem 3 hold for G = Ar(K) or Br(K) and Dr(K), respectively, and ω is not one of the exceptional weights mentioned in Theorem 2 for G = Cr(K), then dim(x − 1)M ≥ (p − 1)N(G, ϕ). Theorems 1 and 2 and the part of Corollary 4 concerning the special linear and symplectic groups were announced in (I.D. Suprunenko, Big Jordan blocks in images of root elements in irreducible representations of the special linear and symplectic groups and estimates for the dimensions of certain subspaces in irreducible modules, in Russian, Doklady NAN Belarusi, 56 (2012), no 1, 36–42).

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On assumptions, restrictions, and tools. Results on the minimal polynomials of elements of order p in representations of simple alge- braic groups in characteristic p (I.D. Suprunenko, Minimal polynomials of elements of order p in irreducible representations of Chevalley groups over fields of characteristic p, Siberian Advances in Mathematics, 6 (1996), 97–150) imply that the image of a nontrivial root element associated with the maximal root in ϕ has no Jordan blocks of size p if l(ϕ) < p − 1. In the following cases l(ϕ) = p−1 and the image of such element has just one block of size p: a) G = Ar(K), ω(ϕ) = a1ω1 + arωr, and a1 + ar = p − 1; b) G = Br(K) or Dr(K) and ω(ϕ) = p−1

2 ω2;

c) G = Cr(K) and ω(ϕ) = (p − 1)ω1. Earlier (1997) it has been proved that for each type of the classical groups there exists a linear function f such that for every unipotent element x of order p the image ϕ(x) has at least f(r)

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Jordan blocks of size p if an irreducible representation ϕ is p-large. We can take f(r) =                            2r − 2 for G = Ar(K), 8r − 10 for G = Br(K), p > 3, 6r − 7 for G = Br(K), p = 3, 4r − 4 for G = Cr(K), 8r − 16 for G = Dr(K), p > 3, 6r − 10 for G = Dr(K), p = 3, 4r − 8 for G = Dr(K), p = 2. For types Ar, Br, and Dr these estimates are asymptotically exact. Let x be a nontrivial root element of G for G = Ar(K) or Dr(K) and a nontrivial long root element for G = Br(K).

• Examples. a) G = Ar(K), ω(ϕ) = a1ω1 + arωr, and a1 + ar = p.

Then ϕ(x) has at most 2r blocks of size p. b) G = Br(K) or Dr(K) and ω(ϕ) = ω1 + p−1

2 ω2.

Then ϕ(x) has at most 8r − 8 blocks of size p for G = Br(K) and p > 3, at most 6r − 5 such blocks for G = Br(K) and p = 3, at most 8r − 12 blocks of size p for G = Dr(K) and p > 3, and at most 6r − 7 such blocks for G = Dr(K) and p = 3. Hence the assertions of Theorems 1 and 3 do not hold for arbitrary p-restricted p-large

• representations. But, may be, some exceptional cases in these theorems can be eliminated.

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Some tools To prove Theorems 1-3, we reduce the problem to root elements for groups of types Ar and Dr and to long root elements for those of types Br and Cr. Lemma 5 Let x and y be unipotent elements of G and x lie in the Zarisky closure of the conjugacy class of y. Then ϕ(x) has no more blocks of size p than ϕ(y) for every irreducible representation ϕ of G. Restrictions to certain subsystem subgroups containing root elements. A subsystem subgroup is a subgroup generated by the root subgroups associated with all roots of a subsystem of the root system of G.

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Let x = 1 be a root element of G for G = Ar(K) and Dr(K) and a long root element for G = Br(K) and Cr(K). Denote by A the relevant root subgroup. Then CG(A) contains a subsystem subgroup S such that S ∼ =            Ar−2(K) for G = Ar(K), A1(K)Br−2(K) for G = Br(K), Cr−1(K) for G = Cr(K), A1(K)Dr−2(K) for G = Dr(K). Put H = AS. For an irreducible p-restricted p-large representation ϕ of G set t = t(ϕ) = l(ϕ) − p + 2 and denote by n(ϕ) the number of Jordan blocks of size p for an element ϕ(x). Analyzing the restriction ϕ|H, we prove that n(ϕ) is not smaller than the sum of the dimen- sions of certain t irreducible S-modules that depend upon ϕ.

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Then we apply Lubeck’s estimates for the dimensions of the majority of irreducible repre- sentations of the classical algebraic groups (F.Lubeck, Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math 4(2001), 135–169). Accord- ing to these results, if s is not too small, then the dimension of an irreducible As(K)-module usually is not smaller than s3/8 and for other classical groups of rank s such dimension usually is not smaller than s3; all exceptions are described. Restrictions on the group rank in Theo- rems 1–3 are caused by using the latter results. If we reduce these restrictions, more exceptions can occur. These results can be easily transferred to irreducible representations of finite classical groups in defining characteristic. They can be useful for recognizing representations and linear groups.

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• II. Restrictions of p-large representations

to subsystem subgroups. The restrictions of p-large representations of classical algebraic groups to subsystem subgroups

• f the maximal rank with two simple components are investigated. Under certain assumptions on

the component ranks it is proved that the restriction of a p-restricted irreducible representation

• f such group in characteristic p to a subsystem subgroup of the form indicated above has a

composition factor equivalent to the tensor product of p-large representations of the components if the value of the highest weight on the maximal root is at least 2p. Theorem 6 Let ϕ be a p-restricted irreducible representation of G, H ⊂ G be a subsys- tem subgroup of type Ak × Ar−k−1 for G = Ar(K), Bk × Dr−k for G = Br(K), Ck × Cr−k for G = Cr(K), and Dk × Dr−k for G = Dr(K); and Hj, j = 1, 2, be the simple components of H. Assume that l(ϕ) ≥ 2p; 2 ≤ k ≤ r − 3 for G = Ar(K), 3 ≤ k ≤ r − 4 for G = Br(K), 2 ≤ k ≤ r − 2 for G = Cr(K), and 4 ≤ k ≤ r − 4 for G = Dr(K). Then the restriction ϕ|H has a factor of the form ϕ1 ⊗ ϕ2 where ϕj is a p-large represen- tation of Hj.

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• Tools. In all cases we explicitly construct a nonzero weight vector that is fixed by all positive

root subgroups of H and the restrictions of whose weights to H1 and H2 are p-large. At this moment it is not clear whether we can demand that both ϕ1 and ϕ2 are p-restricted. The group Ar(K) has an irreducible representation ϕ with l(ϕ) = 2p − 2 whose restriction to H has no composition factors mentioned in Theorem 6. Lemma 7 Let G = Ar(K), the subgroups H and Hj be such as in Theorem 6, and ϕ be an irreducible representation of G with highest weight (p − 1)ω1 + (p − 1)ωr. Then the restriction ϕ|H has no composition factors of the form ϕ1 ⊗ ϕ2 where ϕj is a p-large representation of Hj.

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In general, results on restrictions of representations of simple algebraic groups to subsys- tem subgroups with several simple components can be helpful for investigating restrictions of such representations to simple subsystem subgroups as well (estimates for the multiplicities

• f certain composition factors, etc.). A.E. Zalesskii has drawn my attention to this topic. In

(I.D. Suprunenko and A.E. Zalesskii, On restricting representations of simple algebraic groups to semisimple subgroups with two simple components, Trudy Instituta matematiki, 13 (2005), no. 2, 109–115) it is proved that the restriction of a nontrivial representation of a simple algebraic group to a subsystem subgroup with two simple components almost always has a composition factor that is nontrivial for both components, all exceptions are indicated explicitly. The results on restrictions to subsystem subgroups with two components discussed above can be applied for investigating the behaviour of unipotent elements from proper subsystem subgroups in irreducible representations of the classical groups, first of all, elements that have several nontrivial Jordan blocks in the standard realization of the group. They will permit one to obtain new estimates for the number of Jordan blocks of the maximal size in the images of unipotent elements for different classes of elements and representations.

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