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The linkage bound is 2.

If the algebraic group is simple, say over an algebraically closed $k$, then one has the following lemma.

Lemma. If $V$, $W$ are finite dimensional, there is an $m$ depending on $V$, $W$, so that if $St_n$ is the $n$-th Steinberg module with $n\geq m$ the natural map $Ext^i(V,W)\to Ext^i(V,W\otimes St_n\otimes St_n)$ vanishes.

See for instance my 1977 joint paper with Cline, Parshall and Scott. So this gives a way to kill extension classes while staying within the category of finite dimensional representations. Now it seems one may argue as in Sasha's answer, dualized.

Indeed consider a representative $E$ of an element of $Ext^1(V,W)$. Let us write $W\otimes St_n \otimes St_n$ as $I_n(W)$. For $n$ large the lemma implies that $E$ is the Yoneda composite $P\circ f$ of $P:0\to W\to I_n(W)\to I_n(W)/W\to 0$ with an element $f:V\to I_n(W)/W$. Now if one has a representative of $Ext^i(V,W)$ with $i>1$, write it as Yoneda composite $E\circ F$ of an $E\in Ext^1(Z,W)$ and an $F\in Ext^{i-1}(V,Z)$. Applying the previous result one gets a linkage map from $E\circ F$ towards a Yoneda composite $P\circ Q$ of the representative $P:0\to W\to I_n(W)\to I_n(W)/W\to 0$ with a representative $Q=f\circ F$ of an element of $Ext^{i-1}(V, I_n(W)/W)$. So we use that there is a linkage from $(P\circ f)\circ F$ towards $P\circ(f\circ F)$. One repeats this untill one has a linkage map from the original $E\circ F$ towards a Yoneda composite $R\circ S$ of an extension $R:0\to W\to I_{n_1}(W)\to I_{n_2}(X_1)\to \cdots I_{n_{i-2}}(X_{i-2})\to I_{n_{i-2}}(X_{i-2})/X_{i-2}\to 0$ [ depending only on $W$ ] with an element $S$ of $Ext^1(V,I_n(X_{i-2})/X_{i-2})$. Given a second representative of the same class of $Ext^i(V,W)$ one may repeat the construction and if the $n_i$ are big enough one ends up with the same class of $Ext^1(V,I_n(X_{i-2})/X_{i-2})$, by a dimension shift argument. So the linkage is bounded by 2 as in Sasha's answer.

For reductive groups over an algebraically closed $k$ the Lemma still holds, interpreted appropriately. For nonreductive groups one may take an exhaustive filtration $k\subset M_1\subset M_2\cdots$ of the coordinate ring $k[G]$ by finite dimensional submodules [ for the action by left translation ] and replace $St_n \otimes St_n$ with $M_n$ in the argument above. This is less explicit. It uses that the limit over $n$ of the $Ext^i(V,W\otimes M_n)$ vanishes.

The linkage bound is 2.

If the algebraic group is simple, say over an algebraically closed $k$, then one has the following lemma.

Lemma. If $V$, $W$ are finite dimensional, there is an $m$ depending on $V$, $W$, so that if $St_n$ is the $n$-th Steinberg module with $n\geq m$ the natural map $Ext^i(V,W)\to Ext^i(V,W\otimes St_n\otimes St_n)$ vanishes for $i\geq 1$.

See for instance my 1977 joint paper with Cline, Parshall and Scott. So this gives a way to kill extension classes while staying within the category of finite dimensional representations. Now it seems one may argue as in Sasha's answer, dualized.

Indeed consider a representative $E$ of an element of $Ext^1(V,W)$. Let us write $W\otimes St_n \otimes St_n$ as $I_n(W)$. For $n$ large the lemma implies that $E$ is the Yoneda composite $P\circ f$ of $P:0\to W\to I_n(W)\to I_n(W)/W\to 0$ with an element $f:V\to I_n(W)/W$. Now if one has a representative of $Ext^i(V,W)$ with $i>1$, write it as Yoneda composite $E\circ F$ of an $E\in Ext^1(Z,W)$ and an $F\in Ext^{i-1}(V,Z)$. Applying the previous result one gets a linkage map from $E\circ F$ towards a Yoneda composite $P\circ Q$ of the representative $P:0\to W\to I_n(W)\to I_n(W)/W\to 0$ with a representative $Q=f\circ F$ of an element of $Ext^{i-1}(V, I_n(W)/W)$. So we use that there is a linkage from $(P\circ f)\circ F$ towards $P\circ(f\circ F)$. One repeats this untill one has a linkage map from the original $E\circ F$ towards a Yoneda composite $R\circ S$ of an extension $R:0\to W\to I_{n_1}(W)\to I_{n_2}(X_1)\to \cdots I_{n_{i-2}}(X_{i-2})\to I_{n_{i-2}}(X_{i-2})/X_{i-2}\to 0$ [ depending only on $W$ ] with an element $S$ of $Ext^1(V,I_n(X_{i-2})/X_{i-2})$. Given a second representative of the same class of $Ext^i(V,W)$ one may repeat the construction and if the $n_i$ are big enough [ meaning they grow fast enough ] one ends up with the same class of $Ext^1(V,I_n(X_{i-2})/X_{i-2})$, by a dimension shift argument. So the linkage is bounded by 2 as in Sasha's answer.

For reductive groups over an algebraically closed $k$ the Lemma still holds, interpreted appropriately. For nonreductive groups one may take an exhaustive filtration $k\subset M_1\subset M_2\cdots$ of the coordinate ring $k[G]$ by finite dimensional submodules [ for the action by left translation ] and replace $St_n \otimes St_n$ with $M_n$ in the argument above. The argument needs to be modified a bit. For instance the lemma is no longer available. But the limit over $n$ of the $Ext^i(V,W\otimes M_n)$ vanishes and that is enough to construct all the needed maps.

If the algebraic group is simple, say over an algebraically closed $k$, then one has the following lemma.

Lemma. If $V$, $W$ are finite dimensional, there is an $m$ depending on $V$, $W$, so that if $St_n$ is the $n$-th Steinberg module with $n\geq m$ the natural map $Ext^i(V,W)\to Ext^i(V,W\otimes St_n\otimes St_n)$ vanishes.

See for instance my 1977 joint paper with Cline, Parshall and Scott. So this gives a way to kill extension classes while staying within the category of finite dimensional representations. Now it seems one may argue as in Sasha's answer, dualized.

Indeed consider a representative $E$ of an element of $Ext^1(V,W)$. Let us write $W\otimes St_n \otimes St_n$ as $I_n(W)$. For $n$ large $E$ is the Yoneda composite $P\circ f$ of $P:0\to W\to I_n(W)\to I_n(W)/W\to 0$ with an element $f:V\to I_n(W)/W$. Now if one has a representative of $Ext^i(V,W)$ with $i>1$, write it as Yoneda composite $E\circ F$ of an $E\in Ext^1(Z,W)$ and an $F\in Ext^{i-1}(V,Z)$. Applying the previous result one gets a linkage map from $E\circ F$ towards a Yoneda composite $P\circ Q$ of the representative $P:0\to W\to I_n(W)\to I_n(W)/W\to 0$ with a representative $Q$ of an element of $Ext^{i-1}(V, I_n(W)/W)$. One repeats this untill one has a linkage map towards a Yoneda composite $R\circ S$ of an extension $R:0\to W\to I_{n_1}(W)\to I_{n_1}(X_1)\to \cdots I_{n_{i-2}}(X_{i-2})\to I_{n_{i-2}}(X_{i-2})/X_{i-2}\to 0$ [ depending only on $W$ ] with an element $S$ of $Ext^1(V,I_n(X_{i-2})/X_{i-2})$. Given a second representative of the same class of $Ext^i(V,W)$ one may repeat the construction and if the $n_i$ are big enough one ends up with the same class of $Ext^1(V,I_n(X_{i-2})/X_{i-2})$. So the linkage is bounded by 2 as in Sasha's answer.

For reductive groups over an algebraically closed $k$ the Lemma still holds, interpreted appropriately. For nonreductive groups one may take an exhaustive filtration $k\subset M_1\subset M_2\cdots$ of the coordinate ring $k[G]$ by finite dimensional submodules [ for the action by left translation ] and replace $St_n \otimes St_n$ with $M_n$ in the argument above. This is less explicit. It uses that the limit over $n$ of the $Ext^i(V,W\otimes M_n)$ vanishes.

The linkage bound is 2.

If the algebraic group is simple, say over an algebraically closed $k$, then one has the following lemma.

Lemma. If $V$, $W$ are finite dimensional, there is an $m$ depending on $V$, $W$, so that if $St_n$ is the $n$-th Steinberg module with $n\geq m$ the natural map $Ext^i(V,W)\to Ext^i(V,W\otimes St_n\otimes St_n)$ vanishes.

See for instance my 1977 joint paper with Cline, Parshall and Scott. So this gives a way to kill extension classes while staying within the category of finite dimensional representations. Now it seems one may argue as in Sasha's answer, dualized.

Indeed consider a representative $E$ of an element of $Ext^1(V,W)$. Let us write $W\otimes St_n \otimes St_n$ as $I_n(W)$. For $n$ large the lemma implies that $E$ is the Yoneda composite $P\circ f$ of $P:0\to W\to I_n(W)\to I_n(W)/W\to 0$ with an element $f:V\to I_n(W)/W$. Now if one has a representative of $Ext^i(V,W)$ with $i>1$, write it as Yoneda composite $E\circ F$ of an $E\in Ext^1(Z,W)$ and an $F\in Ext^{i-1}(V,Z)$. Applying the previous result one gets a linkage map from $E\circ F$ towards a Yoneda composite $P\circ Q$ of the representative $P:0\to W\to I_n(W)\to I_n(W)/W\to 0$ with a representative $Q=f\circ F$ of an element of $Ext^{i-1}(V, I_n(W)/W)$. So we use that there is a linkage from $(P\circ f)\circ F$ towards $P\circ(f\circ F)$. One repeats this untill one has a linkage map from the original $E\circ F$ towards a Yoneda composite $R\circ S$ of an extension $R:0\to W\to I_{n_1}(W)\to I_{n_2}(X_1)\to \cdots I_{n_{i-2}}(X_{i-2})\to I_{n_{i-2}}(X_{i-2})/X_{i-2}\to 0$ [ depending only on $W$ ] with an element $S$ of $Ext^1(V,I_n(X_{i-2})/X_{i-2})$. Given a second representative of the same class of $Ext^i(V,W)$ one may repeat the construction and if the $n_i$ are big enough one ends up with the same class of $Ext^1(V,I_n(X_{i-2})/X_{i-2})$, by a dimension shift argument. So the linkage is bounded by 2 as in Sasha's answer.

For reductive groups over an algebraically closed $k$ the Lemma still holds, interpreted appropriately. For nonreductive groups one may take an exhaustive filtration $k\subset M_1\subset M_2\cdots$ of the coordinate ring $k[G]$ by finite dimensional submodules [ for the action by left translation ] and replace $St_n \otimes St_n$ with $M_n$ in the argument above. This is less explicit. It uses that the limit over $n$ of the $Ext^i(V,W\otimes M_n)$ vanishes.

If the algebraic group is simple, say over an algebraically closed $k$, then one has the following lemma.

Lemma. If $V$, $W$ are finite dimensional, there is an $m$ depending on $V$, $W$, so that if $St_n$ is the $n$-th Steinberg module with $n\geq m$ the natural map $Ext^i(V,W)\to Ext^i(V,W\otimes St\otimes St)$ vanishes.

See for instance my 1977 joint paper with Cline, Parshall and Scott. So this gives a way to kill extension classes while staying within the category of finite dimensional representations. Now it seems one may argue as in Sasha's answer, dualized.

Indeed consider a representative $E$ of an element of $Ext^1(V,W)$. Let us write $W\otimes St_n \otimes St_n$ as $I_n(W)$. For $n$ large $E$ is the Yoneda composite $P\circ f$ of $P:0\to W\to I_n(W)\to I_n(W)/W\to 0$ with an element $f:V\to I_n(W)/W$. Now if one has a representative of $Ext^i(V,W)$ with $i>1$, write it as Yoneda composite $E\circ F$ of an $E\in Ext^1(Z,W)$ and an $F\in Ext^{i-1}(V,Z)$. Applying the previous result one gets a linkage map from $E\circ F$ towards a Yoneda composite $P\circ Q$ of the representative $P:0\to W\to I_n(W)\to I_n(W)/W\to 0$ with a representative $Q$ of an element of $Ext^{i-1}(V, I_n(W)/W)$. One repeats this untill one has a linkage map towards a Yoneda composite $R\circ S$ of an extension $R:0\to W\to I_{n_1}(W)\to I_{n_1}(X_1)\to \cdots I_{n_{i-2}}(X_{i-2})\to I_{n_{i-2}}(X_{i-2})/X_{i-2}\to 0$ [ depending only on $W$ ] with an element $S$ of $Ext^1(V,I_n(X_{i-2})/X_{i-2})$. Given a second representative of the same class of $Ext^i(V,W)$ one may repeat the construction and if the $n_i$ are big enough one ends up with the same class of $Ext^1(V,I_n(X_{i-2})/X_{i-2})$. So the linkage is bounded by 2 as in Sasha's answer.

For reductive groups over an algebraically closed $k$ the Lemma still holds, interpreted appropriately. But for nonreductive groups one would need something different.

If the algebraic group is simple, say over an algebraically closed $k$, then one has the following lemma.

Lemma. If $V$, $W$ are finite dimensional, there is an $m$ depending on $V$, $W$, so that if $St_n$ is the $n$-th Steinberg module with $n\geq m$ the natural map $Ext^i(V,W)\to Ext^i(V,W\otimes St_n\otimes St_n)$ vanishes.

See for instance my 1977 joint paper with Cline, Parshall and Scott. So this gives a way to kill extension classes while staying within the category of finite dimensional representations. Now it seems one may argue as in Sasha's answer, dualized.

Indeed consider a representative $E$ of an element of $Ext^1(V,W)$. Let us write $W\otimes St_n \otimes St_n$ as $I_n(W)$. For $n$ large $E$ is the Yoneda composite $P\circ f$ of $P:0\to W\to I_n(W)\to I_n(W)/W\to 0$ with an element $f:V\to I_n(W)/W$. Now if one has a representative of $Ext^i(V,W)$ with $i>1$, write it as Yoneda composite $E\circ F$ of an $E\in Ext^1(Z,W)$ and an $F\in Ext^{i-1}(V,Z)$. Applying the previous result one gets a linkage map from $E\circ F$ towards a Yoneda composite $P\circ Q$ of the representative $P:0\to W\to I_n(W)\to I_n(W)/W\to 0$ with a representative $Q$ of an element of $Ext^{i-1}(V, I_n(W)/W)$. One repeats this untill one has a linkage map towards a Yoneda composite $R\circ S$ of an extension $R:0\to W\to I_{n_1}(W)\to I_{n_1}(X_1)\to \cdots I_{n_{i-2}}(X_{i-2})\to I_{n_{i-2}}(X_{i-2})/X_{i-2}\to 0$ [ depending only on $W$ ] with an element $S$ of $Ext^1(V,I_n(X_{i-2})/X_{i-2})$. Given a second representative of the same class of $Ext^i(V,W)$ one may repeat the construction and if the $n_i$ are big enough one ends up with the same class of $Ext^1(V,I_n(X_{i-2})/X_{i-2})$. So the linkage is bounded by 2 as in Sasha's answer.

For reductive groups over an algebraically closed $k$ the Lemma still holds, interpreted appropriately. For nonreductive groups one may take an exhaustive filtration $k\subset M_1\subset M_2\cdots$ of the coordinate ring $k[G]$ by finite dimensional submodules [ for the action by left translation ] and replace $St_n \otimes St_n$ with $M_n$ in the argument above. This is less explicit. It uses that the limit over $n$ of the $Ext^i(V,W\otimes M_n)$ vanishes.