This is also not a complete answer, but I think technically the right track: Paul Broussous already suggested in his answer to define the Steinberg representation by induction of parabolics and Jim Humphreys pointed out correctly that in general there seems to be no easy branching rules (and suggested a small example, say $n=2$). I just want to tell you only what I see special about your case:

a) To be slightly more explicit on the first point: Let $I$ be a set of simple roots for $G$ $$St:=\sum_{J\subset I} (-1)^{|J|} Ind_{P_J}^{G}(1)$$ and this is related to to the similar expression for a Weyl group character, with $W_J\subset W$ the Weyl group generated by the simple elements longest element of the parabolic subsystem generated by the simple roots $J$: $$\epsilon:=\sum_{J\subset I} (-1)^{|J|} Ind_{W_J}^{W}(1)$$ One shows (Carter "Finite Groups of Lie Type", Chp. 6.2) that $\epsilon(w)=(-1)^{length(w)}$ and $\epsilon(w)=det(w)$ in the reflection representation.

Especially for $SL_{2n}=A_{2n-1}$ all $W_J$ are of type $A_k\times A_{k'}\cdots$ and $\epsilon$ is the sign-function in the symmetric group $\mathbb{S}_{2n}$.

b) A special relation between the Weyl groups / root systems / Tits buildings of $SL_{2n}=A_{2n-1}$ and $Sp_{2n}=C_n$ is as follows:

On $SL_{2n}$ you have an outer automorphism, say explicitly $f(X)=(X^T)^{-1}$. Correspondingly on $A_{2n-1}$ you have a diagram-automorphism $f$ of order $2$ and the orbits form a root system of type $C_n$.

Hence you can

• identify the $J'$-subsets for $Sp_{2n}$ with those $J$-subsets for $SL_{2n}$ which are $f$-stable; and especially the simple roots $\alpha_i',i\in I'$ of $Sp_{2n}$ with $f$-orbits $\{\alpha_i,\alpha_{2n-i}\}$ resp. $\{\alpha_n\}$ in $SL_{2n}$.
• Identify the Weyl group $W'\subset W$ of $Sp_{2n}\subset SL_{2n}$ to be generated by $s_is_{2n-i}$ and $s_n$.
• and so on...I think this is a good embedding of the Tits building in the sense of the comments above?

I am not sure this is enough to compute explicit branching rules for your question $$\chi:=Res^{SL_{2n}}_{Sp_{2n}}(St_{SL_{2n}})=\sum_{J\subset I} (-1)^{|J|} Res^{SL_{2n}}_{Sp_{2n}}(Ind_{P_J}^{SL_{2n}}(1))$$ i.e. to calculate $(\psi,\chi)$ for some irreps $\psi$ of $Sp_{2n}$. You might have a chance by starting with the principal series (maybe it's enough?), proceeding as in many proofs (e.g. $(St,St)=1$ in Carter Prop. 6.2.2), but I don't know an analogon to calculate double cosets $P_{J_1}'xP_{J_2}$ with $P_{J_1}'\subset Sp_{2n}$. At least on the level of Weyl groups computing double cosets $W_{J_1}'xW_{J_2}$ is definitely doable with the description in b), but I think this is not enough...

This is also not a complete answer, but I think technically the right track: Paul Broussous already suggested in his answer to define the Steinberg representation by induction of parabolics and Jim Humphreys pointed out that for general inclusions of Lie-Groupe there seem to be no easy branching rules (and suggested a small example, say $n=2$). I just want to tell you only what I see special about your case:

a) To be slightly more explicit on the first point: Let $I$ be a set of simple roots for $G$ $$St:=\sum_{J\subset I} (-1)^{|J|} Ind_{P_J}^{G}(1)$$ and this is related to to the similar expression for a Weyl group character, with $W_J\subset W$ the Weyl group generated by the simple elements longest element of the parabolic subsystem generated by the simple roots $J$: $$\epsilon:=\sum_{J\subset I} (-1)^{|J|} Ind_{W_J}^{W}(1)$$ One shows (Carter "Finite Groups of Lie Type", Chp. 6.2) that $\epsilon(w)=(-1)^{length(w)}$ and $\epsilon(w)=det(w)$ in the reflection representation.

Especially for $SL_{2n}=A_{2n-1}$ all $W_J$ are of type $A_k\times A_{k'}\cdots$ and $\epsilon$ is the sign-function in the symmetric group $\mathbb{S}_{2n}$.

b) A special relation between the Weyl groups / root systems / Tits buildings of $SL_{2n}=A_{2n-1}$ and $Sp_{2n}=C_n$ is as follows:

On $SL_{2n}$ you have an outer automorphism, say explicitly $f(X)=(X^T)^{-1}$. Correspondingly on $A_{2n-1}$ you have a diagram-automorphism $f$ of order $2$ and the orbits form a root system of type $C_n$.

Hence you can

• identify the $J'$-subsets for $Sp_{2n}$ with those $J$-subsets for $SL_{2n}$ which are $f$-stable; and especially the simple roots $\alpha_i',i\in I'$ of $Sp_{2n}$ with $f$-orbits $\{\alpha_i,\alpha_{2n-i}\}$ resp. $\{\alpha_n\}$ in $SL_{2n}$.
• Identify the Weyl group $W'\subset W$ of $Sp_{2n}\subset SL_{2n}$ to be generated by $s_is_{2n-i}$ and $s_n$.
• and so on...I think this is a good embedding of the Tits building in the sense of the comments above?

I am not sure this is enough to compute explicit branching rules for your question $$\chi:=Res^{SL_{2n}}_{Sp_{2n}}(St_{SL_{2n}})=\sum_{J\subset I} (-1)^{|J|} Res^{SL_{2n}}_{Sp_{2n}}(Ind_{P_J}^{SL_{2n}}(1))$$ i.e. to calculate $(\psi,\chi)$ for some irreps $\psi$ of $Sp_{2n}$. You might have a chance by starting with the principal series (maybe it's enough?), proceeding as in many proofs (e.g. $(St,St)=1$ in Carter Prop. 6.2.2), but I don't know an analogon to calculate double cosets $P_{J_1}'xP_{J_2}$ with $P_{J_1}'\subset Sp_{2n}$. At least on the level of Weyl groups computing double cosets $W_{J_1}'xW_{J_2}$ is definitely doable with the description in b), but I think this is not enough...