# rt.representation theory – The attached representation of the symplectic group in characteristic 2

For a first $$p$$ and something $$g geq 2$$, consider the attached representation $$mathfrak {sp} _ {2g} ( mathbb {F} _p)$$ of the symplectic group $$text {Sp} _ {2g} ( mathbb {F} _p)$$. by $$p geq 3$$It is not difficult to prove that this is an irreducible representation. However, it is reducible in features. $$2$$.

To explain this, we will choose the coordinates. To consider $$text {Sp} _ {2g} ( mathbb {F} _p)$$ as the set of $$2g times 2g$$ block matrices $$M$$ such that
$$M left ( begin {array} {c | c} 0 & 1 \ hline -1 & 0 end {array} right) M ^ t = left ( begin {array} {c | c} 0 & 1 \ hline -1 & 0 end {array} right).$$
With these coordinates,
$$mathfrak {sp} _ {2g} ( mathbb {F} _p) = { text { left ( begin {array} {c | c} A & B \ hline C & -A ^ t end {array} right) | B ^ t = B and C ^ t = C } }.$$
The group $$text {Sp} _ {2g} ( mathbb {F} _p)$$ Acts on this by conjugation. Define
$$V = { text { left ( begin {array} {c | c} A & B \ hline C & -A ^ t end {array} right) in mathfrak {sp } _ {2g} ( mathbb {F} _2) | the diagonal entries of B and C are 0 } }.$$
One can then calculate that the subspace $$V$$ is preserved by $$text {Sp} _ {2g} ( mathbb {F} _2)$$ and that as a representation of $$text {Sp} _ {2g} ( mathbb {F} _2)$$ we have
$$mathfrak {sp} _ {2g} ( mathbb {F} _2) / V cong mathbb {F} _2 ^ {2g}$$
with the obvious action of $$text {Sp} _ {2g} ( mathbb {F} _2)$$.

The quotient map
$$Psi colon mathfrak {sp} _ {2g} ( mathbb {F} _2) longrightarrow mathbb {F} _2 ^ {2g}$$
take an element of $$mathfrak {sp} _ {2g} ( mathbb {F} _2)$$ to the vector whose inputs are the diagonal entries of $$B$$ Y $$C$$. This brings me to my question:

Question: Does anyone know a conceptual explanation for $$Psi$$?

I learned about $$Psi$$ of the classic of Igusa "On the gradual ring of teta-constants", where it is implicit in some of its calculations with the symplectic group. However, it basically falls out of a lot of matrix calculations, and I do not have a deep understanding of why it exists.

(By the way, my previous question here was motivated by trying to understand Igusa's calculations – the great response I got inspired me to follow up!)