The ℓ-modular representation of reductive groups over finite local rings of length two

Theorem 2.1 ([5, Theorem 6.8.3])

Let 𝐺 be a finite group and 𝑁 a normal subgroup of 𝐺. Given a block b ∈ Bl ⁡ ( G ) , we have

1. there exists a block e ∈ Bl ⁡ ( N ) such that b ⋅ e ≠ 0 , so b ∈ Bl ⁡ ( G | e ) .

2. If 𝑒 is as in (1), then b = Tr H G ⁡ ( d ) := ∑ g ∈ G / H g ⁢ d ⁢ g - 1 for a unique block d ∈ Bl ⁡ ( H | e ) , where 𝐻 is the stabilizer of 𝑒 in 𝐺 under the conjugation action.

3. The assignment d ↦ Tr H G ⁡ ( d ) = b gives a bijection between Bl ⁡ ( H | e ) and Bl ⁡ ( G | e ) .

Proposition 2.2 ([5, Theorem 6.8.13])

Let 𝐻 be a finite group, 𝑁 a normal subgroup of 𝐻, and 𝑒 an 𝐻-stable block of defect zero of K ⁢ N . Set S = K ⁢ N ⁢ e , and suppose that 𝐾 is a splitting field for 𝑆. Set L = H / N . For any x ∈ H , there is s x ∈ S * such that x ⁢ t ⁢ x - 1 = s x ⁢ t ⁢ ( s x ) - 1 for all t ∈ S and such that s x ⁢ s y = s x ⁢ y if at least one of x , y is in 𝑁. Then the 2-cocycle α ∈ Z 2 ⁢ ( H ; K * ) defined by s x ⁢ s y = α ⁢ ( x , y ) ⁢ s x ⁢ y for x , y ∈ N depends only on the images of x , y in 𝐿 and induces a 2-cocycle, still denoted 𝛼, in Z 2 ⁢ ( L ; K * ) , and we have an isomorphism of 𝐾-algebras

sending x ⁢ e to s x ⊗ x ¯ , where x ∈ H and x ¯ is the image of 𝑥 in 𝐿.

Lemma 3.1 ([15, Lemma 4.1])

The map β ↦ χ β defines an isomorphism of abelian groups g * → Irr K ⁡ ( g ) that is 𝐺-invariant, i.e. for g ∈ G , we have

The stabilizer H 2 / N and H 2 ′ / N of e χ are isomorphic.

Proof

Since g ⋅ e χ = e g ⋅ χ , the stabilizer of e χ is the same as the stabilizer of 𝜒 (see [7, Section 9]). Thus, we compute the stabilizer of 𝜒. By the discussion above, we can consider 𝜒 to be in Irr ⁡ ( g ) ; by Lemma 3.1, we have that χ = χ β for some β ∈ g * . Thus,

Since the map 𝜎 is an automorphism, it follows that H 2 / N ≅ H 2 ′ / N . ∎

Let α ∈ Z 2 ⁢ ( H ; K * ) be the 2-cocyle associated to the block e χ of K ⁢ N , as in Proposition 2.2. There exists a projective representation of 𝐻 extending 𝜒 which has 𝛼 as its associated cofactor.

Proof

Let 𝑌 be the representation associated to 𝜒, i.e. Y : N → GL n ⁢ ( K ) . Let 𝑅 be a set of representatives of L = H / N in 𝐻 and S = K ⁢ N ⁢ e . For each x ∈ R , by Proposition 2.2, there is some s x ∈ S * such that x ⁢ t ⁢ x - 1 = s x ⁢ t ⁢ ( s x ) - 1 for all t ∈ S . Note also, since 𝑒 is an 𝐻-stable block of defect zero of K ⁢ N , then we have S = K ⁢ N ⁢ e ≅ M n ⁢ ( K ) . With abuse of notation, we can think of s x as an element of GL n ⁢ ( K ) . Now define 𝑋 to be the map X : H → GL n ⁢ ( K ) such that, for each h = x ⁢ n ∈ H , we have X ⁢ ( h ) = s x ⁢ Y ⁢ ( n ) with x ∈ R and n ∈ N . One can check that 𝑋 as defined above is a projective representation of 𝐻 with factor set 𝛼 in Z 2 ⁢ ( L ; K * ) that extends 𝑌. ∎

Let 𝐺 be a finite group and 𝑀 an F α ⁢ G -module with α ∈ Z 2 ⁢ ( G ; R * ) . Then 𝑀 contains a lattice 𝐿 that is an R α ⁢ G -module.

Proof

Note, by [9, Lemma 2.2.1], to show that 𝐿 is a lattice of 𝑀, it is enough to show that 𝐿 is finitely generated as an 𝑅-module and 𝐿 generates 𝑀 as an 𝐹-vector space. Thus, pick an 𝐹 basis e 1 , … , e n of 𝑀; then let

be a lattice of 𝑀. Let L := ∑ g ∈ G g ⁢ L ′ ; then 𝐿 is an R α ⁢ G -module. Note that 𝐿 is finitely generated as an 𝑅-module by { g ⁢ e i : 1 ≤ i ≤ n , g ∈ G } , and it also generates 𝑀 as a vector space. Thus, 𝐿 is a 𝐺-invariant lattice. ∎

Proposition 4.3 ([17, Theorem 9.6.1])

Let ( F ; R ; K ) be a splitting 𝑝-modular system in which 𝑅 is complete, and let 𝐺 be a group of order p d ⁢ q , where 𝑞 is prime to 𝑝. Let 𝑇 be an F ⁢ G -module of dimension 𝑛, containing a full R ⁢ G -sublattice T 0 . The following are equivalent:

1. p d | n and 𝑇 is a simple F ⁢ G -module.

2. The homomorphism R ⁢ G → End R ⁢ ( T 0 ) that gives the action of R ⁢ G on T 0 identifies End R ⁢ ( T 0 ) ≅ M n ⁢ ( R ) with a ring direct summand of R ⁢ G .

3. 𝑇 is a simple F ⁢ G -module, and T 0 is a projective R ⁢ G -module.

4. The homomorphism K ⁢ G → End K ⁢ ( T 0 / π ⁢ T 0 ) identifies

   End K ⁢ ( T 0 / π ⁢ T 0 ) ≅ M n ⁢ ( K )

   with a ring direct summand of K ⁢ G .

5. As a K ⁢ G -module, T 0 / π ⁢ T 0 is simple and projective.

Given an 𝐻-stable block e χ of defect zero of K ⁢ N , where N ⁢ ⊴ ⁢ H , let 𝑉 be the unique ordinary simple module associated to this block. According to Proposition 4.1, let V ^ be the projective representation of 𝐻 that extends 𝑉, with cofactor α ^ in Z 2 ⁢ ( H / N ; R * ) . Any 𝐻-invariant lattice of V ^ , call it 𝐿, gives rise to a K α ⁢ H -module that extends the simple projective K ⁢ N -module associated to e χ , where α = α ^ mod ( π ) .

Proof

Let V ^ be as above, so V ^ is an F α ^ ⁢ H -module such that Res N H ⁡ ( V ^ ) ≃ V as an F ⁢ N -module. Now, by Lemma 4.2, take 𝐿 to be an 𝐻-invariant lattice of V ^ ; then consider its reduction to a K α ⁢ H -module, call it L ¯ := K ⊗ R L . Note that 𝐿 is also an 𝑁-invariant lattice of 𝑉, the unique simple ordinary module associated to the block e χ . Thus, by Proposition 4.3, the reduction of 𝐿 is a simple module of K ⁢ N . Therefore, Res N H ⁡ ( L ¯ ) is isomorphic to the simple module associated to this block. One can conclude that L ¯ is a K α ⁢ H -module that extends the simple projective K ⁢ N -module associated to e χ , where α = α ^ mod ( π ) . ∎

The 𝛼 obtained from Theorem 4.4 is trivial.

Proof

Note, by work of Stasinski and Vera-Gajardo, it was proven that any 𝜒 element of Irr F ⁡ ( N ) extends to its stabilizer 𝐻 (see [15, Proposition 4.5]). Thus, by Proposition 4.1, the cofactor associated to the extension of 𝜒 is trivial over 𝐹, i.e. α ^ = 1 . By Theorem 4.4, there is a K α ⁢ H -module L ¯ that extends the simple projective module associated to e χ , where α = α ^ mod ( π ) = 1 . Thus, 𝛼 is trivial. ∎

There exists an isomorphism of block algebras between e ⁢ K ⁢ H 2 and e ⁢ K ⁢ H 2 ′ , where e = e χ for 𝜒 in Irr K ⁡ ( N ) and H 2 and H 2 ′ are the stabilizer of 𝑒 in G 2 and G 2 ′ respectively.

Proof

We can apply Proposition 2.2 since 𝑒 is 𝐻-stable block of defect zero of K ⁢ N . Thus, we have an isomorphism of 𝐾-algebras

since, by Proposition 4.5, α - 1 is trivial. ∎

Lemma 5.1 ([16, Lemma 16.1])

Let 𝐻 be a subgroup of 𝐺 of index n, and 𝐵 an interior 𝐻-algebra. Then we have Ind H G ⁢ ( B ) ≅ M n ⁢ ( B ) as 𝐾-algebras.

Proposition 5.2 ([16, Proposition 16.6])

Let 𝐴 be an interior 𝐺-algebra, and let 𝐻 be a subgroup of 𝐺. Assume that there exists an idempotent i ∈ A H such that 1 A = Tr H G ⁡ ( i ) and i g ⁢ i = 0 for all g ∈ G - H . Then there is an isomorphism of interior 𝐺-algebras

given by x ⊗ b ⊗ y ↦ x ⋅ b ⋅ y ( x , y ∈ G , b ∈ i ⁢ A ⁢ i ).

Given a block 𝑏 of K ⁢ G 2 , there is a block Φ ^ ⁢ ( b ) of K ⁢ G 2 ′ such that the 𝐾-algebras b ⁢ K ⁢ G 2 and Φ ^ ⁢ ( b ) ⁢ K ⁢ G 2 ′ are isomorphic.

Proof

Fix χ ∈ Irr ⁡ ( N ) up to conjugation by 𝐺, and let e = e χ be a primitive idempotent of K ⁢ N ; now fix a block b ∈ Bl ⁡ ( G 2 | e ) such that, by Clifford’s theorem (Theorem 2.1), we have b = Tr H 2 G 2 ⁡ ( d ) for some primitive idempotent 𝑑 of K ⁢ H 2 , where H 2 is the stabilizer of 𝑒 in G 2 . Note that, by Theorem 4.6, we have that Φ : e ⁢ K ⁢ H 2 ≅ e ⁢ K ⁢ H 2 ′ as 𝐾-algebras. Now the map Φ gives a bijection between Bl ⁡ ( H 2 | e ) and Bl ⁡ ( H 2 ′ | e ) such that Φ : d ⁢ K ⁢ H 2 ≅ Φ ⁢ ( d ) ⁢ K ⁢ H 2 ′ for each d ∈ Bl ⁡ ( H 2 | e ) . Moreover, Theorem 2.1 tells us there is a bijection between Bl ⁡ ( G | e ) and Bl ⁡ ( H | e ) . Thus, we obtain a bijection between Bl ⁡ ( G 2 | e ) and Bl ⁡ ( G 2 ′ | e ) by the map

Let A = b ⁢ K ⁢ G 2 , so 1 A = b = Tr H 2 G 2 ⁡ ( d ) . Note that d ⋅ b = d , so d ∈ A H 2 and d g ⁢ d = 0 for g ∈ G 2 - H 2 (see [5, proof of Lemma 6.8.4]). Thus, we have that

Thus, by Proposition 5.2, there is an isomorphism of interior G 2 -algebras:

Note by Lemma 5.1 that Ind H 2 G 2 ⁢ ( d ⁢ K ⁢ H 2 ) ≅ M n ⁢ ( d ⁢ K ⁢ H 2 ) as 𝐾-algebras, with n = | G 2 : H 2 | = | G 2 ′ : H 2 ′ | . Since Φ : d ⁢ K ⁢ H 2 ≅ Φ ⁢ ( d ) ⁢ K ⁢ H 2 ′ , we conclude that

Thus, we have that b ⁢ K ⁢ G 2 and Φ ^ ⁢ ( b ) ⁢ K ⁢ G 2 ′ are isomorphic as 𝐾-algebras. ∎

Let G 2 = G ⁢ ( O 2 ) and G 2 ′ = G ⁢ ( O 2 ′ ) be the group of points of any reductive group scheme 𝔾 over ℤ such that 𝑝 is very good for G × F q or G = GL n . There exists an isomorphism of group algebra K ⁢ G ⁢ ( O 2 ) ≅ K ⁢ G ⁢ ( O 2 ′ ) , where 𝐾 is a sufficiently large field of characteristic different from 𝑝.

Proof

Define the map Ψ : K ⁢ G 2 → K ⁢ G 2 ′ by

First, we show that Ψ is an algebra homomorphism. Note that, by definition, it is a linear map since each of the Φ ^ from Proposition 5.3 are. Now, given 𝑥 and 𝑦 both in K ⁢ G 2 , we want to show that Ψ ⁢ ( x ⁢ y ) = Ψ ⁢ ( x ) ⁢ Ψ ⁢ ( y ) . To prove this, it is sufficient to show that different blocks of G 2 are mapped to different blocks of G 2 ′ . Because if they are, we have that

since the products of elements of different blocks are zero. Moreover, by Φ ^ being a 𝐾-algebra homomorphism for each block 𝑏, we have

Now, to show that different blocks of G 2 are mapped to different blocks of G 2 ′ , we look at two cases. Let 𝑏 and 𝑐 be two different blocks of G 2 .

Case 1: Assume 𝑏 and 𝑐 cover the same block 𝑒 of K ⁢ N . Thus, by Clifford’s theorem (Theorem 2.1), b = Tr H 2 G 2 ⁡ ( d 1 ) and c = Tr H 2 G 2 ⁡ ( d 2 ) such that d 1 ⋅ d 2 = 0 . Recall by Theorem 4.6 that the map Φ : e ⁢ K ⁢ H 2 ≅ e ⁢ K ⁢ H 2 ′ is a 𝐾-algebra isomorphism, and thus Φ ⁢ ( d 1 ) ⁢ Φ ⁢ ( d 2 ) = 0 . Applying Theorem 2.1 for the blocks of K ⁢ G 2 ′ over 𝑒, we have that

since Φ ⁢ ( d 1 ) ≠ Φ ⁢ ( d 2 ) . Thus, Φ ^ ⁢ ( b ) ≠ Φ ^ ⁢ ( c ) .

Case 2: Assume 𝑏 and 𝑐 cover different blocks e 1 and e 2 of K ⁢ N respectively. Note, to prove that Φ ^ ⁢ ( b ) ≠ Φ ^ ⁢ ( c ) , it is enough to show that they cover different blocks of K ⁢ N as blocks of G 2 ′ . Assume the opposite. By definition of the map Φ ^ , we also have that Φ ^ ⁢ ( b ) and Φ ^ ⁢ ( c ) cover the blocks e 1 and e 2 of K ⁢ N respectively. Thus, by [5, Proposition 6.8.2], we have that e 1 and e 2 are G 2 ′ -conjugated blocks. Since K ⁢ N is a semisimple algebra, i.e. all the blocks have defect zero, e 1 and e 2 are defined by unique irreducible characters 𝜒 and 𝜏 of K ⁢ N . Thus, e 1 and e 2 are G 2 ′ -conjugated blocks if and only if 𝜒 and 𝜏 are G 2 ′ -conjugated characters. If χ g = τ for some g ∈ G 2 ′ , then for any h ∈ G 2 such that h ¯ = σ ⁢ ( g ¯ ) , we have χ h = τ , where the bars denote reductions to the residue field F q . Thus, if 𝜒 and 𝜏 are G 2 ′ -conjugated characters, then 𝜒 and 𝜏 are also G 2 -conjugated. Thus, e 1 and e 2 are G 2 -conjugated. This is a contradiction with the fact that 𝑏 and 𝑐 cover different blocks e 1 and e 2 of K ⁢ N .

Therefore, we can conclude that different blocks of G 2 are mapped to different blocks of G 2 ′ . Hence, we have an algebra homomorphism from Ψ : K ⁢ G 2 → K ⁢ G 2 ′ , and since each Φ ^ is injective and maps different blocks to different blocks, we can conclude Ψ is an injective map. Thus, by dimension reasons, Ψ is also surjective. Therefore, K ⁢ G 2 ≅ K ⁢ G 2 ′ as 𝐾-algebras. ∎