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!)