Modules for Algebraic Groups with Finitely Many Orbits on Singular 1

Home , Algebraic group, Composition series, Linear algebraic group

Modules for algebraic groups with ﬁnitely many orbits on singular 1-spaces

Aluna Rizzoli

Imperial College London

Groups, representations and applications: new perspectives I P denotes a parabolic subgroup, B a Borel subgroup.

I We use λi to denote a fundamental dominant weight with respect to a root system Φ = hαi ii .

I If G has root system Φ, VG (λi ) is then the irreducible rational G-module of highest weight λi .

Linear algebraic groups

We work with algebraic groups over an algebraically closed ﬁeld k of characteristic p ≥ 0.

I We let Tn denote an n-dimensional torus. I We use λi to denote a fundamental dominant weight with respect to a root system Φ = hαi ii .

I If G has root system Φ, VG (λi ) is then the irreducible rational G-module of highest weight λi .

Linear algebraic groups

We work with algebraic groups over an algebraically closed ﬁeld k of characteristic p ≥ 0.

I We let Tn denote an n-dimensional torus. I P denotes a parabolic subgroup, B a Borel subgroup. I If G has root system Φ, VG (λi ) is then the irreducible rational G-module of highest weight λi .

Linear algebraic groups

We work with algebraic groups over an algebraically closed ﬁeld k of characteristic p ≥ 0.

I We let Tn denote an n-dimensional torus. I P denotes a parabolic subgroup, B a Borel subgroup.

I We use λi to denote a fundamental dominant weight with respect to a root system Φ = hαi ii . Linear algebraic groups

We work with algebraic groups over an algebraically closed ﬁeld k of characteristic p ≥ 0.

I We let Tn denote an n-dimensional torus. I P denotes a parabolic subgroup, B a Borel subgroup.

I We use λi to denote a fundamental dominant weight with respect to a root system Φ = hαi ii .

I If G has root system Φ, VG (λi ) is then the irreducible rational G-module of highest weight λi . A double coset problem

We now describe the motivating problem behind our research. Let G be a simple connected algebraic group over k. Throughout let H, K be closed connected subgroups of G. The set of (H, K)-double cosets in G is given by

H\G/K := {HgK : g ∈ G}.

Question: When is |H\G/K| < ∞? A double coset problem

H\G/K := {HgK : g ∈ G}.

Question: When is |H\G/K| < ∞? A double coset problem

H\G/K := {HgK : g ∈ G}.

Question: When is |H\G/K| < ∞? A double coset problem

H\G/K := {HgK : g ∈ G}.

Question: When is |H\G/K| < ∞? A double coset problem

H\G/K := {HgK : g ∈ G}.

Question: When is |H\G/K| < ∞? A double coset problem: some examples

I |B\G/B| < ∞ (or |P\G/P| < ∞) for any Borel (or parabolic) subgroup of G (Bruhat decomposition).

I G = SL(V ).

I G = HK (classiﬁed in [Liebeck et al., 1996]) is equivalent to |H\G/K| = 1. For example SO7 = G2P1 or SO8 = B3P1,

where V ↓ B3 = VB3 (λ3). I If H, K are both reductive and |H\G/K| < ∞ then by [Brundan, 2000] there is actually only a single double coset. In characteristic 0 this is a consequence of Luna’s slice theorem. Hence to complete the |H\G/K| ﬁniteness problem (with H, K maximal), we can take K parabolic. A double coset problem: some examples

I |B\G/B| < ∞ (or |P\G/P| < ∞) for any Borel (or parabolic) subgroup of G (Bruhat decomposition).

I G = SL(V ).

I G = HK (classiﬁed in [Liebeck et al., 1996]) is equivalent to |H\G/K| = 1. For example SO7 = G2P1 or SO8 = B3P1,

I |B\G/B| < ∞ (or |P\G/P| < ∞) for any Borel (or parabolic) subgroup of G (Bruhat decomposition).

I G = SL(V ).

I G = HK (classiﬁed in [Liebeck et al., 1996]) is equivalent to |H\G/K| = 1. For example SO7 = G2P1 or SO8 = B3P1,

I |B\G/B| < ∞ (or |P\G/P| < ∞) for any Borel (or parabolic) subgroup of G (Bruhat decomposition).

I G = SL(V ).

I G = HK (classiﬁed in [Liebeck et al., 1996]) is equivalent to |H\G/K| = 1. For example SO7 = G2P1 or SO8 = B3P1,

I |B\G/B| < ∞ (or |P\G/P| < ∞) for any Borel (or parabolic) subgroup of G (Bruhat decomposition).

I G = SL(V ).

I G = HK (classiﬁed in [Liebeck et al., 1996]) is equivalent to |H\G/K| = 1. For example SO7 = G2P1 or SO8 = B3P1,

I |H\G/K| < ∞ if and only if H has ﬁnitely many orbits on m-spaces.

I [Guralnick et al., 1997] ﬁnds all irreducible modules for semisimple algebraic groups with ﬁnitely many orbits on m-spaces.

I We call modules with ﬁnitely many orbits on P1(V ) ﬁnite orbit modules.

Solved for SLn

When G = SLn the problem has been solved by [Guralnick et al., 1997]. In fact: I |H\G/K| < ∞ if and only if H has ﬁnitely many orbits on m-spaces.

I [Guralnick et al., 1997] ﬁnds all irreducible modules for semisimple algebraic groups with ﬁnitely many orbits on m-spaces.

I We call modules with ﬁnitely many orbits on P1(V ) ﬁnite orbit modules.

Solved for SLn

When G = SLn the problem has been solved by [Guralnick et al., 1997]. In fact:

I In SLn maximal parabolic subgroups K = Pm correspond to stabilizers of m-spaces. I [Guralnick et al., 1997] ﬁnds all irreducible modules for semisimple algebraic groups with ﬁnitely many orbits on m-spaces.

I We call modules with ﬁnitely many orbits on P1(V ) ﬁnite orbit modules.

Solved for SLn

When G = SLn the problem has been solved by [Guralnick et al., 1997]. In fact:

I In SLn maximal parabolic subgroups K = Pm correspond to stabilizers of m-spaces.

I |H\G/K| < ∞ if and only if H has finitely many orbits on m-spaces. I We call modules with finitely many orbits on P1(V ) finite orbit modules.

Solved for SLn

When G = SLn the problem has been solved by [Guralnick et al., 1997]. In fact:

I In SLn maximal parabolic subgroups K = Pm correspond to stabilizers of m-spaces.

I |H\G/K| < ∞ if and only if H has ﬁnitely many orbits on m-spaces.

I [Guralnick et al., 1997] ﬁnds all irreducible modules for semisimple algebraic groups with ﬁnitely many orbits on m-spaces. Solved for SLn

When G = SLn the problem has been solved by [Guralnick et al., 1997]. In fact:

I In SLn maximal parabolic subgroups K = Pm correspond to stabilizers of m-spaces.

I |H\G/K| < ∞ if and only if H has ﬁnitely many orbits on m-spaces.

I [Guralnick et al., 1997] ﬁnds all irreducible modules for semisimple algebraic groups with ﬁnitely many orbits on m-spaces.

I We call modules with finitely many orbits on P1(V ) finite orbit modules. Finite orbit modules Theorem [Guralnick et al., 1997] Let G be a connected simple algebraic group over k = k. If V is an irreducible rational finite orbit kG-module then either V is an internal module for G or V is one of the following: GV dim V p i i 2 An λ1 + p λ1, λ1 + p λn (n + 1) 6= 0 A2 λ1 + λ2 7 p = 3 A3 λ1 + λ2 16 p = 3 B4 λ4 16 any B5 λ5 32 any C3 λ2 13 p = 3 G2 λ1 7 − δp,2 any F4 λ4 25 p = 3 Table: Finite orbit modules Internal modules

One way to obtain ﬁnite orbit modules is the following:

I Let H be a simple algebraic group and P = QG be a proper parabolic subgroup of H with unipotent radical Q and Levi subgroup G.

I Let 1 = Q0 < Q1 < ··· < Qr = Q be a G-invariant composition series for Q.

I Then each factor Qi+1/Qi has the structure of a rational irreducible kG-module.

I The Levi subgroup G has only ﬁnitely many orbits on each module Qi+1/Qi . Internal modules

One way to obtain ﬁnite orbit modules is the following:

I Let H be a simple algebraic group and P = QG be a proper parabolic subgroup of H with unipotent radical Q and Levi subgroup G.

I Let 1 = Q0 < Q1 < ··· < Qr = Q be a G-invariant composition series for Q.

I Then each factor Qi+1/Qi has the structure of a rational irreducible kG-module.

I The Levi subgroup G has only ﬁnitely many orbits on each module Qi+1/Qi . Internal modules

One way to obtain ﬁnite orbit modules is the following:

I Let H be a simple algebraic group and P = QG be a proper parabolic subgroup of H with unipotent radical Q and Levi subgroup G.

I Let 1 = Q0 < Q1 < ··· < Qr = Q be a G-invariant composition series for Q.

I Then each factor Qi+1/Qi has the structure of a rational irreducible kG-module.

I The Levi subgroup G has only ﬁnitely many orbits on each module Qi+1/Qi . Internal modules

One way to obtain ﬁnite orbit modules is the following:

I Let H be a simple algebraic group and P = QG be a proper parabolic subgroup of H with unipotent radical Q and Levi subgroup G.

I Let 1 = Q0 < Q1 < ··· < Qr = Q be a G-invariant composition series for Q.

I Then each factor Qi+1/Qi has the structure of a rational irreducible kG-module.

I The Levi subgroup G has only ﬁnitely many orbits on each module Qi+1/Qi . Internal modules

One way to obtain ﬁnite orbit modules is the following:

I Let H be a simple algebraic group and P = QG be a proper parabolic subgroup of H with unipotent radical Q and Levi subgroup G.

I Let 1 = Q0 < Q1 < ··· < Qr = Q be a G-invariant composition series for Q.

I Then each factor Qi+1/Qi has the structure of a rational irreducible kG-module.

I The Levi subgroup G has only ﬁnitely many orbits on each module Qi+1/Qi . I The number of G-orbits on 1-spaces is equal to 2 plus the number of orbits of D5 on 2-spaces.

I D5 has 6 orbits on 2-spaces in the half-spin module. (Upcoming work).

I V = VA1 (λ1) ⊗ VD5 (λ4).

I Here VA1 (λ1) is the natural 2-dimensional module for A1 and

VD5 (λ5) is the 32-dimensional half-spin module for D5. As discussed G has ﬁnitely many orbits on 1-spaces in V .

Internal module: an example

Let us consider the 64-dimensional internal module V for G = A1D5 ≤ E7.

I A1D5 is a Levi factor of a P6-parabolic in E7. I The number of G-orbits on 1-spaces is equal to 2 plus the number of orbits of D5 on 2-spaces.

I D5 has 6 orbits on 2-spaces in the half-spin module. (Upcoming work).

I Here VA1 (λ1) is the natural 2-dimensional module for A1 and

VD5 (λ5) is the 32-dimensional half-spin module for D5. As discussed G has ﬁnitely many orbits on 1-spaces in V .

Internal module: an example

Let us consider the 64-dimensional internal module V for G = A1D5 ≤ E7.

I A1D5 is a Levi factor of a P6-parabolic in E7.

I V = VA1 (λ1) ⊗ VD5 (λ4). I The number of G-orbits on 1-spaces is equal to 2 plus the number of orbits of D5 on 2-spaces.

I D5 has 6 orbits on 2-spaces in the half-spin module. (Upcoming work).

As discussed G has ﬁnitely many orbits on 1-spaces in V .

Internal module: an example

Let us consider the 64-dimensional internal module V for G = A1D5 ≤ E7.

I A1D5 is a Levi factor of a P6-parabolic in E7.

I V = VA1 (λ1) ⊗ VD5 (λ4).

I Here VA1 (λ1) is the natural 2-dimensional module for A1 and

VD5 (λ5) is the 32-dimensional half-spin module for D5. I The number of G-orbits on 1-spaces is equal to 2 plus the number of orbits of D5 on 2-spaces.

I D5 has 6 orbits on 2-spaces in the half-spin module. (Upcoming work).

Internal module: an example

Let us consider the 64-dimensional internal module V for G = A1D5 ≤ E7.

I A1D5 is a Levi factor of a P6-parabolic in E7.

I V = VA1 (λ1) ⊗ VD5 (λ4).

I Here VA1 (λ1) is the natural 2-dimensional module for A1 and

VD5 (λ5) is the 32-dimensional half-spin module for D5. As discussed G has ﬁnitely many orbits on 1-spaces in V . I D5 has 6 orbits on 2-spaces in the half-spin module. (Upcoming work).

Internal module: an example

Let us consider the 64-dimensional internal module V for G = A1D5 ≤ E7.

I A1D5 is a Levi factor of a P6-parabolic in E7.

I V = VA1 (λ1) ⊗ VD5 (λ4).

I Here VA1 (λ1) is the natural 2-dimensional module for A1 and

VD5 (λ5) is the 32-dimensional half-spin module for D5. As discussed G has ﬁnitely many orbits on 1-spaces in V .

I The number of G-orbits on 1-spaces is equal to 2 plus the number of orbits of D5 on 2-spaces. Internal module: an example

Let us consider the 64-dimensional internal module V for G = A1D5 ≤ E7.

I A1D5 is a Levi factor of a P6-parabolic in E7.

I V = VA1 (λ1) ⊗ VD5 (λ4).

I Here VA1 (λ1) is the natural 2-dimensional module for A1 and

VD5 (λ5) is the 32-dimensional half-spin module for D5. As discussed G has ﬁnitely many orbits on 1-spaces in V .

I The number of G-orbits on 1-spaces is equal to 2 plus the number of orbits of D5 on 2-spaces.

I D5 has 6 orbits on 2-spaces in the half-spin module. (Upcoming work). Our setting and theorem We consider the double coset problem when G = SOn and K = P1 is a maximal parabolic subgroup of G stabilizing a singular 1-space. Theorem [AR, 2019] Let H be a simple irreducible closed connected subgroup of G = SO(V ) such that H has finitely many orbits on singular 1-spaces. Then either V is a finite orbit module or a composition factor of the adjoint module, or up to field or graph twists (H, V ) is as follows. HV dim V p Stabilizer of dense orbit

A1 4λ1 5 ≥ 5 Alt4 0 0 B6 λ6 64 2 P1P1 < G2G2 3 3 C3 λ2 14 6= 3 (A1) .(2 .3) 4 4 C4 λ2 26 2 (A1) .(2 .Alt4) F4 λ4 26 p 6= 3 D4.3 Table: Finite singular orbit modules for simple groups Our setting and theorem We consider the double coset problem when G = SOn and K = P1 is a maximal parabolic subgroup of G stabilizing a singular 1-space. Theorem [AR, 2019] Let H be a simple irreducible closed connected subgroup of G = SO(V ) such that H has finitely many orbits on singular 1-spaces. Then either V is a finite orbit module or a composition factor of the adjoint module, or up to field or graph twists (H, V ) is as follows. HV dim V p Stabilizer of dense orbit

A1 4λ1 5 ≥ 5 Alt4 0 0 B6 λ6 64 2 P1P1 < G2G2 3 3 C3 λ2 14 6= 3 (A1) .(2 .3) 4 4 C4 λ2 26 2 (A1) .(2 .Alt4) F4 λ4 26 p 6= 3 D4.3 Table: Finite singular orbit modules for simple groups I V is not a ﬁnite orbit module since dim A1 = 3 < dim V − 1 = dim P1(V ). I V is an orthogonal module.

I A quadratic form Q on V can be obtained by setting Q(v1 ⊗ v2 ⊗ v3 ⊗ v4, u1 ⊗ u2 ⊗ u3 ⊗ u4) = (v1, u1)(v2, u2)(v3, u3)(v4, u4). I A point in a dense H-orbit on singular 1-spaces in V must have a ﬁnite stabilizer.

Some considerations:

The A1 case: setup

Let p > 5 and let H = SL2(k). Let W = he, f i be the natural 2-dimensional H-module, where e, f is a hyperbolic pair for an alternating form on W stabilized by H. Then V = S4(W ) is an irreducible 5-dimensional module for H = A1. I V is not a ﬁnite orbit module since dim A1 = 3 < dim V − 1 = dim P1(V ). I V is an orthogonal module.

Some considerations:

The A1 case: setup

I V is not a ﬁnite orbit module since dim A1 = 3 < dim V − 1 = dim P1(V ). I A quadratic form Q on V can be obtained by setting Q(v1 ⊗ v2 ⊗ v3 ⊗ v4, u1 ⊗ u2 ⊗ u3 ⊗ u4) = (v1, u1)(v2, u2)(v3, u3)(v4, u4). I A point in a dense H-orbit on singular 1-spaces in V must have a ﬁnite stabilizer.

The A1 case: setup

I V is not a ﬁnite orbit module since dim A1 = 3 < dim V − 1 = dim P1(V ). I V is an orthogonal module. I A point in a dense H-orbit on singular 1-spaces in V must have a ﬁnite stabilizer.

The A1 case: setup

I V is not a ﬁnite orbit module since dim A1 = 3 < dim V − 1 = dim P1(V ). I V is an orthogonal module.

I A quadratic form Q on V can be obtained by setting Q(v1 ⊗ v2 ⊗ v3 ⊗ v4, u1 ⊗ u2 ⊗ u3 ⊗ u4) = (v1, u1)(v2, u2)(v3, u3)(v4, u4). The A1 case: setup

I V is not a ﬁnite orbit module since dim A1 = 3 < dim V − 1 = dim P1(V ). I V is an orthogonal module.

The A1 case: the orbits

We can easily ﬁnd two orbits:

I he ⊗ e ⊗ e ⊗ ei is ﬁxed by a P1 parabolic stabilising hei.

I he ⊗ e ⊗ e ⊗ f + e ⊗ e ⊗ f ⊗ e + e ⊗ f ⊗ e ⊗ e + f ⊗ e ⊗ e ⊗ ei is ﬁxed only by a T1. Note that both 1-spaces are singular.

The A1 case: the orbits

We can easily ﬁnd two orbits:

I he ⊗ e ⊗ e ⊗ ei is ﬁxed by a P1 parabolic stabilising hei.

I he ⊗ e ⊗ e ⊗ f + e ⊗ e ⊗ f ⊗ e + e ⊗ f ⊗ e ⊗ e + f ⊗ e ⊗ e ⊗ ei is ﬁxed only by a T1. Note that both 1-spaces are singular.

The A1 case: the orbits

We can easily ﬁnd two orbits:

I he ⊗ e ⊗ e ⊗ ei is ﬁxed by a P1 parabolic stabilising hei.

I he ⊗ e ⊗ e ⊗ f + e ⊗ e ⊗ f ⊗ e + e ⊗ f ⊗ e ⊗ e + f ⊗ e ⊗ e ⊗ ei is ﬁxed only by a T1. The A1 case: the orbits

We can easily ﬁnd two orbits:

I he ⊗ e ⊗ e ⊗ ei is ﬁxed by a P1 parabolic stabilising hei.

I he ⊗ e ⊗ e ⊗ f + e ⊗ e ⊗ f ⊗ e + e ⊗ f ⊗ e ⊗ e + f ⊗ e ⊗ e ⊗ ei is ﬁxed only by a T1.

Note that both 1-spaces are singular. I The dense orbit splits into 4 orbits if q ≡ 1 mod 3 and into 2-orbits otherwise. The stabilizers have size 3, 3, 4, 12 and 2, 2 respectively.

I Adding up the sizes of the orbits we have found gives us the number of singular 1-spaces, therefore 6 is a uniform bound on the number of A1-orbits.

Going to ﬁnite ﬁelds and using the Lang-Steinberg theorem we prove:

The A1 case: the dense orbit

We can assume that k = Fq. By standard theory there is a subgroup of H isomorphic to Alt4.

I Alt4 ﬁxes two singular 1-spaces in V . This can be seen using standard character theory.

I Alt4 is in fact the full stabilizer of both 1-spaces. This follows from the subgroup structure of PGL2(q). I The dense orbit splits into 4 orbits if q ≡ 1 mod 3 and into 2-orbits otherwise. The stabilizers have size 3, 3, 4, 12 and 2, 2 respectively.

I Adding up the sizes of the orbits we have found gives us the number of singular 1-spaces, therefore 6 is a uniform bound on the number of A1-orbits.

Going to ﬁnite ﬁelds and using the Lang-Steinberg theorem we prove:

The A1 case: the dense orbit

We can assume that k = Fq. By standard theory there is a subgroup of H isomorphic to Alt4.

I Alt4 ﬁxes two singular 1-spaces in V . This can be seen using standard character theory.

I Adding up the sizes of the orbits we have found gives us the number of singular 1-spaces, therefore 6 is a uniform bound on the number of A1-orbits.

Going to ﬁnite ﬁelds and using the Lang-Steinberg theorem we prove:

The A1 case: the dense orbit

We can assume that k = Fq. By standard theory there is a subgroup of H isomorphic to Alt4.

I Alt4 ﬁxes two singular 1-spaces in V . This can be seen using standard character theory.

I Alt4 is in fact the full stabilizer of both 1-spaces. This follows from the subgroup structure of PGL2(q). I Adding up the sizes of the orbits we have found gives us the number of singular 1-spaces, therefore 6 is a uniform bound on the number of A1-orbits.

The A1 case: the dense orbit

We can assume that k = Fq. By standard theory there is a subgroup of H isomorphic to Alt4.

I Alt4 ﬁxes two singular 1-spaces in V . This can be seen using standard character theory.

I Alt4 is in fact the full stabilizer of both 1-spaces. This follows from the subgroup structure of PGL2(q). Going to ﬁnite ﬁelds and using the Lang-Steinberg theorem we prove:

I The dense orbit splits into 4 orbits if q ≡ 1 mod 3 and into 2-orbits otherwise. The stabilizers have size 3, 3, 4, 12 and 2, 2 respectively. The A1 case: the dense orbit

We can assume that k = Fq. By standard theory there is a subgroup of H isomorphic to Alt4.

I Alt4 ﬁxes two singular 1-spaces in V . This can be seen using standard character theory.

I Alt4 is in fact the full stabilizer of both 1-spaces. This follows from the subgroup structure of PGL2(q). Going to ﬁnite ﬁelds and using the Lang-Steinberg theorem we prove:

I The dense orbit splits into 4 orbits if q ≡ 1 mod 3 and into 2-orbits otherwise. The stabilizers have size 3, 3, 4, 12 and 2, 2 respectively.

I Adding up the sizes of the orbits we have found gives us the number of singular 1-spaces, therefore 6 is a uniform bound on the number of A1-orbits. I VB6 (λ6) is the unique irreducible rational module for B6 of dimension 64.

I It can be obtained by embedding B6 in the Cliﬀord algebra of the natural 14-dimensional module for D7 and looking at the action on spinors.

I p must be 2 for the module to be orthogonal.

We now consider the spin module for B6.

A spin module Our theorem had the following list: HV dim V p Stabilizer of dense orbit

A1 4λ1 5 ≥ 5 Alt4 0 0 B6 λ6 64 2 P1P1 < G2G2 3 3 C3 λ2 14 6= 3 (A1) .(2 .3)(p = 2) 4 4 C4 λ2 26 2 (A1) .(2 .Alt4) F4 λ4 26 p 6= 3 D4.3 Table: Finite singular orbit modules for simple groups I It can be obtained by embedding B6 in the Cliﬀord algebra of the natural 14-dimensional module for D7 and looking at the action on spinors.

I p must be 2 for the module to be orthogonal.

A spin module Our theorem had the following list: HV dim V p Stabilizer of dense orbit

A1 4λ1 5 ≥ 5 Alt4 0 0 B6 λ6 64 2 P1P1 < G2G2 3 3 C3 λ2 14 6= 3 (A1) .(2 .3)(p = 2) 4 4 C4 λ2 26 2 (A1) .(2 .Alt4) F4 λ4 26 p 6= 3 D4.3 Table: Finite singular orbit modules for simple groups

We now consider the spin module for B6.

I VB6 (λ6) is the unique irreducible rational module for B6 of dimension 64. I p must be 2 for the module to be orthogonal.

A spin module Our theorem had the following list: HV dim V p Stabilizer of dense orbit

A1 4λ1 5 ≥ 5 Alt4 0 0 B6 λ6 64 2 P1P1 < G2G2 3 3 C3 λ2 14 6= 3 (A1) .(2 .3)(p = 2) 4 4 C4 λ2 26 2 (A1) .(2 .Alt4) F4 λ4 26 p 6= 3 D4.3 Table: Finite singular orbit modules for simple groups

We now consider the spin module for B6.

I VB6 (λ6) is the unique irreducible rational module for B6 of dimension 64.

I It can be obtained by embedding B6 in the Cliﬀord algebra of the natural 14-dimensional module for D7 and looking at the action on spinors. A spin module Our theorem had the following list: HV dim V p Stabilizer of dense orbit

A1 4λ1 5 ≥ 5 Alt4 0 0 B6 λ6 64 2 P1P1 < G2G2 3 3 C3 λ2 14 6= 3 (A1) .(2 .3)(p = 2) 4 4 C4 λ2 26 2 (A1) .(2 .Alt4) F4 λ4 26 p 6= 3 D4.3 Table: Finite singular orbit modules for simple groups

We now consider the spin module for B6.

I VB6 (λ6) is the unique irreducible rational module for B6 of dimension 64.

I It can be obtained by embedding B6 in the Cliﬀord algebra of the natural 14-dimensional module for D7 and looking at the action on spinors.

I p must be 2 for the module to be orthogonal. Spin module for D7

Using previous work by [Popov, 1980] and [Igusa, 1970] we have the following list of spinor representatives for the D7-orbits on the spin module:

◦ Orbit type Spinor x (D7)x

2 1 U21SL7 3 1 + f1f2f3f4 U27(SL3 × Spin7) 4 1 + f1f2f3f4f5f6 U12SL6 5 1 + f1f2f3f7 + f1f2f3f4f5f6 U21(SL3 × SL3) ∗ 6 λ(1 + f1f2f3f7 + f4f5f6f7 + f1f2f3f4f5f6): λ ∈ k G2 × G2 7 1 + f1f2f3f7 + f1f5f6f7 + f1f2f3f4f5f6 U19(SL2 × Sp4) 8 1 + f1f2f3f7 + f1f5f6f7 + f2f4f6f7 + f1f2f3f4f5f6 U14G2 9 1 + f1f2f3f4 + f3f4f5f6 U26(Sp6 × T1) 10 1 + f1f2f3f4 + f3f4f5f6 + f1f3f6f7 U26SL4 Table: Subgroups of stabilizers if p = 2 Spin module: restriction to B6 To restrict to B6 we need to ﬁnd the orbits of the stabilizers listed on non-singular 1-spaces. 0 0 S Rep v Sv

U21.SL7 e1 + f1 U21.SL6 U26.(Sp6T1) e1 + f1 U25.(Sp4T1) e7 + f7 U14.Sp6 U27.(Spin7SL3) e5 + f5 U23.(Spin7SL2) e1 + f1 U21.(G2SL3) U12.SL6 e1 + f1 U11.SL5 −1 αe7 + α f7 : α 6= 0 SL6 U21.(SL3SL3) e1 + f1 U16.(SL2SL3) e4 + f4 U16.(SL2SL3) −1 αe7 + α f7 : α 6= 0 U15.(SL3SL3) e1 + f1 + f4 U18.(SL2SL2) −1 G2G2 αe7 + α f7 : α 6= 0, 1 (SL3.2)(SL3.2) e7 + f7 G2 × G2 0 f4 + e4 G2P1 0 e1 + f1 P1G2 0 0 e1 + f1 + e4 P1P1 U19.(SL2Sp4) e1 + f1 U10.Sp4 e5 + f5 U16.(SL2SL2) −1 αe4 + α f4 : α 6= 0 U9.(Sp4T1.2) U14.G2 e1 + f1 U14.SL2 −1 αe7 + α f7 : α 6= 0 U8.(SL3.2) U26.SL4 e1 + f1 U22.Sp4 e7 + f7 U20.SL3 Adjoint modules

One of the possibilities in our theorem is that we have a composition factor of the adjoint module. The adjoint module for a connected linear algebraic group H is the module Lie(H) on which H acts by conjugation.

I For example if H = SLn then the adjoint module for H is the Lie-algebra sln of trace-0 matrices. I If p | n then this is not an irreducible module, because scalar matrices are in sln.

I If p | n the non-trivial composition factor sln/hI i is an irreducible module of highest weight λ1 + λn−1. Adjoint modules

I For example if H = SLn then the adjoint module for H is the Lie-algebra sln of trace-0 matrices. I If p | n then this is not an irreducible module, because scalar matrices are in sln.

I If p | n the non-trivial composition factor sln/hI i is an irreducible module of highest weight λ1 + λn−1. Adjoint modules

I For example if H = SLn then the adjoint module for H is the Lie-algebra sln of trace-0 matrices. I If p | n then this is not an irreducible module, because scalar matrices are in sln.

I If p | n the non-trivial composition factor sln/hI i is an irreducible module of highest weight λ1 + λn−1. Adjoint modules

I For example if H = SLn then the adjoint module for H is the Lie-algebra sln of trace-0 matrices. I If p | n then this is not an irreducible module, because scalar matrices are in sln.

I If p | n the non-trivial composition factor sln/hI i is an irreducible module of highest weight λ1 + λn−1. Adjoint modules

I For example if H = SLn then the adjoint module for H is the Lie-algebra sln of trace-0 matrices. I If p | n then this is not an irreducible module, because scalar matrices are in sln.

I If p | n the non-trivial composition factor sln/hI i is an irreducible module of highest weight λ1 + λn−1. I The bound dim V ≤ dim H + 2 is always satisﬁed since dim(Lie(H)) = dim H. 0 I If v, v ∈ V0 are in the same H-orbit, then they are in the same W -orbit.

I If dim V0 ≥ 3, then V is not a ﬁnite singular orbit module.

This implies the following:

Adjoint modules: the zero-space

Let V be a non-trivial composition factor of the adjoint module for a connected simple algebraic group H.

Let V0 ≤ V be the ﬁxed space of a maximal torus T ≤ H. Let W = Weyl(H). I If dim V0 ≥ 3, then V is not a ﬁnite singular orbit module.

0 I If v, v ∈ V0 are in the same H-orbit, then they are in the same W -orbit. This implies the following:

Adjoint modules: the zero-space

Let V be a non-trivial composition factor of the adjoint module for a connected simple algebraic group H.

Let V0 ≤ V be the ﬁxed space of a maximal torus T ≤ H. Let W = Weyl(H).

I The bound dim V ≤ dim H + 2 is always satisﬁed since dim(Lie(H)) = dim H. I If dim V0 ≥ 3, then V is not a ﬁnite singular orbit module.

This implies the following:

Adjoint modules: the zero-space

Let V be a non-trivial composition factor of the adjoint module for a connected simple algebraic group H.

Let V0 ≤ V be the ﬁxed space of a maximal torus T ≤ H. Let W = Weyl(H).

I The bound dim V ≤ dim H + 2 is always satisﬁed since dim(Lie(H)) = dim H. 0 I If v, v ∈ V0 are in the same H-orbit, then they are in the same W -orbit. Adjoint modules: the zero-space

Let V be a non-trivial composition factor of the adjoint module for a connected simple algebraic group H.

Let V0 ≤ V be the ﬁxed space of a maximal torus T ≤ H. Let W = Weyl(H).

I The bound dim V ≤ dim H + 2 is always satisﬁed since dim(Lie(H)) = dim H. 0 I If v, v ∈ V0 are in the same H-orbit, then they are in the same W -orbit. This implies the following:

I If dim V0 ≥ 3, then V is not a ﬁnite singular orbit module. I V = sl4/hI i, where sl4 is the Lie algebra of trace-0 4 × 4 matrices, on which H acts by conjugation.

I V0 is the 2-dimensional subspace of diagonal matrices in V .

I Every singular vector in V is either semisimple or nilpotent.

Considering elements in Jordan Canonical Form shows:

Since there is only 1 orbit on semisimple singular 1-spaces and ﬁnitely many on nilpotent 1-spaces, this concludes.

Adjoint modules, an easy example

Let H = SL4 and V = V (λ1 + λ3) be a composition factor of the adjoint module for H when p = 2. I Every singular vector in V is either semisimple or nilpotent.

I V0 is the 2-dimensional subspace of diagonal matrices in V . Considering elements in Jordan Canonical Form shows:

Since there is only 1 orbit on semisimple singular 1-spaces and ﬁnitely many on nilpotent 1-spaces, this concludes.

Adjoint modules, an easy example

Let H = SL4 and V = V (λ1 + λ3) be a composition factor of the adjoint module for H when p = 2.

I V = sl4/hI i, where sl4 is the Lie algebra of trace-0 4 × 4 matrices, on which H acts by conjugation. I Every singular vector in V is either semisimple or nilpotent.

Considering elements in Jordan Canonical Form shows:

Since there is only 1 orbit on semisimple singular 1-spaces and ﬁnitely many on nilpotent 1-spaces, this concludes.

Adjoint modules, an easy example

Let H = SL4 and V = V (λ1 + λ3) be a composition factor of the adjoint module for H when p = 2.

I V = sl4/hI i, where sl4 is the Lie algebra of trace-0 4 × 4 matrices, on which H acts by conjugation.

I V0 is the 2-dimensional subspace of diagonal matrices in V . I Every singular vector in V is either semisimple or nilpotent.

Since there is only 1 orbit on semisimple singular 1-spaces and ﬁnitely many on nilpotent 1-spaces, this concludes.

Adjoint modules, an easy example

Let H = SL4 and V = V (λ1 + λ3) be a composition factor of the adjoint module for H when p = 2.

I V = sl4/hI i, where sl4 is the Lie algebra of trace-0 4 × 4 matrices, on which H acts by conjugation.

I V0 is the 2-dimensional subspace of diagonal matrices in V . Considering elements in Jordan Canonical Form shows: Since there is only 1 orbit on semisimple singular 1-spaces and ﬁnitely many on nilpotent 1-spaces, this concludes.

Adjoint modules, an easy example

Let H = SL4 and V = V (λ1 + λ3) be a composition factor of the adjoint module for H when p = 2.

I V = sl4/hI i, where sl4 is the Lie algebra of trace-0 4 × 4 matrices, on which H acts by conjugation.

I V0 is the 2-dimensional subspace of diagonal matrices in V . Considering elements in Jordan Canonical Form shows:

I Every singular vector in V is either semisimple or nilpotent. Adjoint modules, an easy example

Let H = SL4 and V = V (λ1 + λ3) be a composition factor of the adjoint module for H when p = 2.

I V = sl4/hI i, where sl4 is the Lie algebra of trace-0 4 × 4 matrices, on which H acts by conjugation.

I V0 is the 2-dimensional subspace of diagonal matrices in V . Considering elements in Jordan Canonical Form shows:

I Every singular vector in V is either semisimple or nilpotent. Since there is only 1 orbit on semisimple singular 1-spaces and ﬁnitely many on nilpotent 1-spaces, this concludes. For a partial result, If H ≤ SO(V ) is maximal semisimple connected and V is an irreducible ﬁnite singular orbit H-module then (H, V ) is one of the following:

HV dim V p Stabilizer of dense orbit

C2C2 λ1 ⊗ λ1 16 p 6= 2 (A1A1).2 p = 2 U3A1 C2Cn, n > 2 λ1 ⊗ λ1 8n p 6= 2 (A1A1).2(Cn−2) p = 2 U3A1(Cn−2) Table: Finite singular orbit modules for maximal semisimple groups

Semisimple case

In order to obtain a complete answer to our double coset problem when G = SO(V ) and K is a P1 parabolic subgroup, we need to know what happens if H is a semisimple group. HV dim V p Stabilizer of dense orbit

C2C2 λ1 ⊗ λ1 16 p 6= 2 (A1A1).2 p = 2 U3A1 C2Cn, n > 2 λ1 ⊗ λ1 8n p 6= 2 (A1A1).2(Cn−2) p = 2 U3A1(Cn−2) Table: Finite singular orbit modules for maximal semisimple groups

Semisimple case

In order to obtain a complete answer to our double coset problem when G = SO(V ) and K is a P1 parabolic subgroup, we need to know what happens if H is a semisimple group. For a partial result, If H ≤ SO(V ) is maximal semisimple connected and V is an irreducible ﬁnite singular orbit H-module then (H, V ) is one of the following: Semisimple case

HV dim V p Stabilizer of dense orbit

C2C2 λ1 ⊗ λ1 16 p 6= 2 (A1A1).2 p = 2 U3A1 C2Cn, n > 2 λ1 ⊗ λ1 8n p 6= 2 (A1A1).2(Cn−2) p = 2 U3A1(Cn−2) Table: Finite singular orbit modules for maximal semisimple groups I Determine modules with ﬁnitely many orbits on totally singular m-spaces.

I Remove maximality requirement on the semisimple case.

I G exceptional.

Upcoming work

Eventually we want to achieve a characterization of ﬁnite singular orbit modules analogous to the one for ﬁnite orbit modules. In order to do this we need to: Upcoming work

Eventually we want to achieve a characterization of ﬁnite singular orbit modules analogous to the one for ﬁnite orbit modules. In order to do this we need to:

I Determine modules with ﬁnitely many orbits on totally singular m-spaces.

I Remove maximality requirement on the semisimple case.

I G exceptional. ReferencesI

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