4.6 General Linear Groups 4.7 Steinbergs Unipotent Characters Farid - PowerPoint PPT Presentation

4.6 General Linear Groups 4.7 Steinberg’s Unipotent Characters Farid Aliniaeifard York University <http://math.yorku.ca/~faridanf/> July 3, 2015

Overview 4.6 General Linear Groups 4.7 Steinberg’s Unipotent Characters

Recall ◮ G n = GL n = GL n ( F q ) and A = A ( G ∗ ) := A ( GL ) ◮ A ( GL ) is a PSH with PSH basis Σ and following Product and coproduct m := Ind GL i + j Infl P i , j P i , j GL i × GL j � K i , j � Res GL i + j ∆ := ( − ) P i , j where � g i � l g i ∈ GL i , g j ∈ GL j , l ∈ F i × j P i , j = q 0 g j � I i � l K i , j = 0 I j

4.6 General linear Groups ◮ Goal: Finding the cardinality of C = Σ ∩ p , where Σ is the PSH basis for A ( GL ) and p is the set of primitive elements of A ( GL ). Why C is important? Any PSH A has a canonical tensor product Theorem 3.12: decomposition � A = A ( ρ ) ρ ∈C with A ( ρ ) a PSH, and ρ the only primitive element in its PSH basis Σ( ρ ) = { σ ∈ Σ : there exsits n ≥ 1 s . t ( σ n , ρ ) � = 0 } . ◮ Also we know from Theorem 3.18 A ( ρ ) ∼ = Λ.

Definition: ◮ Call an irreducible representation ρ of GL n cuspidal for n ≥ 1 if it lies in C . ◮ Given an irreducible character σ of GL n , say that d ( σ ) = n . ◮ Let C n := { ρ ∈ C : d ( ρ ) = n } for n ≥ 1 denote the subset of cuspidal characters of GL n . ◮ Let F denote the set of all nonconstant monic irreducible polynomials f ( x ) � = x in F q [ x ] with nonzero constant term.

Proposition 4.40 The number |C n | of cuspidal characters of GL n ( F q ) is the number of |F n | of irreducible monic degree n polynomials f ( x ) � = x in F q [ x ] with nonzero constant term. Proof: Using induction on n : ◮ For n = 1, GL 1 ( F q ) = F ∗ q and F 1 = { f ( x ) = x − c : c ∈ F ∗ q } ◮ The number of complex characters of GL n ( F q ) = The number of conjugacy classes of GL n ( F q ) ◮ These conjugacy classes are uniquely represented by rational canonical forms, which are parametrized by functions λ : F → Par with the property that � f ∈F deg ( f ) | λ ( f ) | = n . ◮ (4.32) tells us that | Σ n | is similarly parametrized by the functions λ : C → Par having the property that � ρ ∈C deg ( ρ ) | λ ( ρ ) | = n .

F = ⊔ n ≥ 1 F n and C = ⊔ n ≥ 1 C n We have for an equality for all n ≥ 1 �   � � �  λ  � � �  C → Par : deg ( ρ ) | λ ( ρ ) | = n = | Σ n | � � � �  ρ ∈C � � �   � � �  λ  � � � =  F → Par : deg ( f ) | λ ( f ) | = n � � � �  ρ ∈F � � Since there is only one partition λ having | λ | = 1: �   � � �  λ   ⊔ n − 1 � � � |C n | = | Σ n | − i =1 C i → Par : deg ( ρ ) | λ ( ρ ) | = n � � � �  ρ ∈C � � �   � � �  λ   ⊔ n − 1 � � � |F n | = | Σ n | − i =1 F i → Par : deg ( f ) | λ ( f ) | = n � � � �  ρ ∈F � �

Example 4.41 The sets F n of monic irreducible polynomials f ( x ) � = x in F 2 [ x ] of degree n for n ≤ 3, so that we know how many cuspidal characters of GL n ( F q ) in C n to expect: F 1 = { x + 1 } F 2 = { x 2 + x + 1 } F 3 = { x 3 + x + 1 , x 3 + x 2 + 1 } Thus we expect ◮ one cuspidal character of GL 1 ( F 2 ), namely ρ 1 (= 1 GL 1 ( F 2 ) ◮ one cuspidal character ρ 2 of GL 2 ( F 2 ) ′ ◮ two cuspidal character ρ 3 , ρ 3 of GL 3 ( F 2 ).

Steinberg’s Unipotent Characters Define i := 1 GL 1 of GL 1 ( F q ) Let P (1 n ) = B . We have i n = Ind GL n 1 B = C [ GL n / B ] for some n . B Definition: An irreducible character σ of GL n appearing as a constituent of Ind GL n 1 B = C [ GL n / B ] is called a unipotent character. B ◮ In particular, 1 GL n is a unipotent character of GL n for each n . Proposition: One can choose Λ ∼ = A ( GL )( i ) in Theorem 3.20(g) so that h n �→ 1 GL n . Proof: ◮ Theorem 3.18(a) ⇒ i 2 = Ind GL 2 1 B = 1 GL 2 + St 2 . B ◮ Choose the isomorphism so as to send h 2 → 1 GL 2 .

◮ Claim: St ⊥ 2 (1 GL n ) = 0 for every n ≥ 2. � K i , j = � � Res G n � Ex 4 . 25( e ) ⇒ ∆(1 GL n ) = P i , j 1 GL n 1 GL i ⊗ 1 GL j i + j = n i + j = n so that St ⊥ 2 (1 GL n ) = ( St 2 , 1 GL 2 )1 GL n − 2 = 0 since St 2 � = 1 GL 2 ◮ Then h n �→ 1 GL n .

Steinberg’s Unipotent Characters ◮ This subalgebra Λ ∼ = A ( GL )( i ) and the unipotent characters χ λ p corresponding under this isomorphism to the Schur functions s λ was introduced by Steinberg. ◮ He wrote down χ λ p by using Jacobi-Trudi determinant expression for s λ = det ( h λ i − i + j ). ◮ The Steinberg’s unipotent characters St n , which is the unipotent character corresponding under the isomorphism in last Proposition to e n = S (1 n ) , can be defined by the virtual sum St n := χ (1 n ) � ( − 1) n − l ( α ) Ind GL n = P α 1 P α q α in which sum run through all compositions α of n .

References Grinberg and Reiner Hopf Algerba in Combinatorics Bruce Sagan (2000) The Symmetric Group

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