18.704: Seminar in Algebra and Number Theory
Oleg Shamovsky
04/26/05
General parabolic subgroups of Sp2n
Let V be a 2n-dimensional vector space over a field F, B be a non-degenerate symplectic form
of V, Sp2n be the set g ∈ GLV ∣ Bgv, gw  Bv, w v, w ∈ V, S 1 be an arbitrary
isotropic subspace of V, and dimS 1   k 1 .
For r ∈ ℤ  , we arbitrarily pick subspaces S 2 , S 3 , . . . , S r such that
S r ⊂ S r−1 ⊂ . . . ⊂ S 2 ⊂ S 1
Since, S 1 is isotropic, it follows that each S i 2 ≤ i ≤ r is also isotropic. We see that this is
ture because since S 1 is isotropic, Bv, w  0 for all v, w ∈ S 1 . And, since each S i 2 ≤ i ≤ r
is a subspace of S 1 then Bv ′ , w ′   0 for all v ′ , w ′ ∈ S i , thereby making S i isotropic.
Let dimS i   k i 2 ≤ i ≤ r. Then, we know that
k r ≤ k r−1 ≤ . . . ≤ k 2 ≤ k 1
We now define S i to be the orthogonal complement of S i 1 ≤ i ≤ r. Therefore, it is clear that
S r ⊂ S r−1 ⊂ . . . ⊂ S 2 ⊂ S 1 ⊂ S 1 ⊂ S 2 ⊂ . . . ⊂ S r
Let W  S 1 /S 1 to be a set of cosets of the form v  S 1 where v ∈ S 1 . We know that
dimW  dimS 1 /S 1   dimS 1  − dimS 1 . Since dimS 1   k 1 , we also know that
dimS 1   dimV − k 1  2n − k 1 . Let k  k 1 . Therefore
dimW  dimS 1 /S 1 
 dimS 1  − dimS 1 
 2n − k − k
 2n − k
A stabilizer P for S r ⊂ S r−1 ⊂ . . . ⊂ S 2 ⊂ S 1 is defined as
PS r ⊂ . . . ⊂ S 1   g ∈ SpV ∣ gS r  S r , . . . , gS 1  S 1 
To give a homomorphism f from a group P to a direct product group
G r  G r−1  . . .  G 1  G 0 is the same as giving r  1 separate homomorphisms f i : P  G i
where i  0, 1, . . . , r. The relationship is:
fp  f r p, f r−1 p, . . . , f 0 p  G r  . . .  G 0
We now define the following group homomorphism:
f : PS r ⊂ . . . ⊂ S 1   GLS r   GLS r−1 /S r   . . .  GLS 1 /S 2   SpW
as a coordinate homomorphism:
If g ∈ P, then
f r g  g| S r ∈ GLS r 
f r−1 g  g| S r−1 /S r ∈ GLS r−1 /S r 

f 1 g  g| S 1 /S 2 ∈ GLS 1 /S 2 
f 0 g  g| S 1 /S 1 ∈ SpS 1 /S 1 
Note that we use the notation g| A to mean the restriction of g to the subspace A.
Let N f be the kernel of f. We know that the kernel of a group homomorphism is a normal
subgroup. Thus, N f is a normal subgroup of PS r ⊂ . . . ⊂ S 1 .
Let N be a normal subgroup of S r ⊂ . . . ⊂ S 1 (denoted NS r ⊂ . . . ⊂ S 1 ) which is defined
as:
g ∈ SpV such that
gx r   x r x r ∈ S r ,
gx r−1   x r−1  an element of S r  x r−1 ∈ S r−1 ,

gx 1   x 1  an element of S 2  x 1 ∈ S 1 
gw  w  an element of S 1  w ∈ S 1 
With the help of the class, I outlined some examples in lecture showing that the above
definition defines a normal subgroup of PS r ⊂ . . . ⊂ S 1 . Furthermore, from the definition
of f and N f above, we see that N f  NS r ⊂ . . . ⊂ S 1 .
It then follows that f defines an inclusion:
PS r ⊂ . . . ⊂ S 1 /NS r ⊂ . . . ⊂ S 1   GLS r   GLS r−1 /S r   . . .  GLS 1 /S 2   SpW
which is also surjective. Note: the proof of surjectivity is left out of these notes; it involves
finding a number of elements of P and showing that every element of
GLS r   GLS r−1 /S r   . . .  GLS 1 /S 2   SpW is an image (under f) of elements of P/N.
Therefore, f is a bijective map from PS r ⊂ . . . ⊂ S 1 /NS r ⊂ . . . ⊂ S 1  to
GLS r   GLS r−1 /S r   . . .  GLS 1 /S 2   SpW, which implies that
PS r ⊂ . . . ⊂ S 1 /NS r ⊂ . . . ⊂ S 1  ≅ GLS r   GLS r−1 /S r   . . .  GLS 1 /S 2   SpW
Note the difference between this result and the one demonstrated for maximal parabolic
subgroups of Sp2n in a previous talk. Specifically, for maximal parabolic subgroups:
PS/N  GLk, F  SW
whereas for general parabolic subgroups, we have
PS r ⊂ . . . ⊂ S 1 /NS r ⊂ . . . ⊂ S 1  ≅ GLS r   GLS r−1 /S r   . . .  GLS 1 /S 2   SpW