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

Irreducible representations of 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. 1

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 p j λ j with p -restricted λ j . Set µ = � t j =0 λ j (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 p th power. For every positive integer j and rational irreducible representation ϕ of G the images of the representations ϕ and ϕ j = ϕ · Fr j coincide, the canonical Jordan form of the images of any unipotent element in ϕ and ϕ j is the same. So there is no sense to call ϕ j p -large if ϕ is not p -large. If ω is the highest weight of ϕ , then p j ω is the highest weight of ϕ j . That is why we pass to ω . 2

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 obtained. 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 of G , ω i and α i , 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 � = A r ( K ). If ω ( ϕ ) = � r i =1 a i ω i , it is well known that  � r i =1 a i for G = A r ( K ) or C r ( K ),    l ( ϕ ) = a 1 + 2( a 2 + . . . + a r − 1 ) + a r for G = B r ( K ),  a 1 + 2( a 2 + . . . + a r − 2 ) + a r − 1 + a r for G = D r ( K ).   In Theorems 1–3 and Corollary 4 ϕ is a p -restricted irreducible representation of G with highest weight ω = � r i =1 a i ω i . For G = A r ( K ) set ω ∗ = � r i =1 a r +1 − i ω i (this is the highest weight of the representation dual to ϕ ). 3

Theorem 1 Let G = A r ( K ) , r > 8 , and l ( ϕ ) ≥ p . 1) . Assume that l ( ϕ ) ≥ p + 2 , or � r − 2 i =3 a i � = 0 , or a 2 + a r − 1 > 1 , or a 2 + a r − 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) ω 1 + ω 2 + ω r , ( p − 2) ω 1 + 2 ω 2 , a 1 ω 1 + a r ω r , a 1 + a r = p + 2 } . If ω satisfies the assumptions of Item 1 , ω and ω ∗ �∈ Ω , and, furthermore, both ω and ω ∗ � = 2 ω 1 + ω 3 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 = C r ( K ) , r > 12 , and l ( ϕ ) ≥ p . Assume that ω � = ( p − 1) ω 1 + ω 2 . 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) ω 1 + 2 ω 2 and for p = 3 the weight ω � = 2 ω 1 + ω 3 as well, then ϕ ( x ) has at least ( l ( ϕ ) − p + 2)( r − 1) 3 such blocks. Theorem 3 Let p > 2 , G = B r ( K ) or D r ( K ) , r ≥ 12 for G = B r ( K ) and r ≥ 14 for G = D r ( K ) . Assume that l ( ϕ ) ≥ p . 1 . Suppose that � r i =4 a i � = 0 , or a 3 > 1 , or l ( ϕ ) > p and a 2 a 3 � = 0 , or l ( ϕ ) > p + 1 and a 2 > 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) ω 1 + ω 4 , ( p − 1 2 ) ω 2 + ω 3 , ( p − 2) ω 1 + ω 2 + ω 3 , ω 1 + ( p +1 2 ) ω 2 ; a 1 ω 1 + 3 ω 2 , a 1 ≥ p − 3 } . If ω satisfies the assumptions of Item 1 , ω �∈ Ω and, furthermore, ω � = ω 1 + ω 4 for p = 3 , then ϕ ( x ) has at least 2( l ( ϕ ) − p + 2)( r − 2) 3 such blocks. 4

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 = A r ( K ), 2( r − 2) 3 for G = B r ( K ) or D r ( K ), and ( r − 1) 3 for G = C r ( 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 , or Theorem 3 for G = A r ( K ) , C r ( K ) , or B r ( K ) and D r ( 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 = A r ( K ) or B r ( K ) and D r ( K ) , respectively, and ω is not one of the exceptional weights mentioned in Theorem 2 for G = C r ( 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). 5

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 = A r ( K ), ω ( ϕ ) = a 1 ω 1 + a r ω r , and a 1 + a r = p − 1; b) G = B r ( K ) or D r ( K ) and ω ( ϕ ) = p − 1 2 ω 2 ; c) G = C r ( K ) and ω ( ϕ ) = ( p − 1) ω 1 . 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 ) 6

Jordan blocks of size p if an irreducible representation ϕ is p -large. We can take  2 r − 2 for G = A r ( K ),     8 r − 10 for G = B r ( K ), p > 3,      6 r − 7 for G = B r ( K ), p = 3,     f ( r ) = 4 r − 4 for G = C r ( K ),  8 r − 16 for G = D r ( K ), p > 3,      6 r − 10 for G = D r ( K ), p = 3,       4 r − 8 for G = D r ( K ), p = 2.  For types A r , B r , and D r these estimates are asymptotically exact. Let x be a nontrivial root element of G for G = A r ( K ) or D r ( K ) and a nontrivial long root element for G = B r ( K ). Examples. a) G = A r ( K ), ω ( ϕ ) = a 1 ω 1 + a r ω r , and a 1 + a r = p . Then ϕ ( x ) has at most 2 r blocks of size p . b) G = B r ( K ) or D r ( K ) and ω ( ϕ ) = ω 1 + p − 1 2 ω 2 . Then ϕ ( x ) has at most 8 r − 8 blocks of size p for G = B r ( K ) and p > 3, at most 6 r − 5 such blocks for G = B r ( K ) and p = 3, at most 8 r − 12 blocks of size p for G = D r ( K ) and p > 3, and at most 6 r − 7 such blocks for G = D r ( 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. 7

Some tools To prove Theorems 1-3, we reduce the problem to root elements for groups of types A r and D r and to long root elements for those of types B r and C r . 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 . 8