Let $\mathbf{G}$ be a connected, gs-split $\mathrm{\mathbb{k}}$-supergroup; we assume for it
that its tangent Lie superalgebra $\U0001d524:=Lie(\mathbf{G})$ be a $\mathrm{\mathbb{k}}$-form of a complex Lie superalgebra ${\U0001d524}_{\u2102}$ – i.e., there exists a Lie superalgebra ${\U0001d524}_{\mathbb{Z}}$ over $\mathbb{Z}$ such that $\U0001d524=\mathrm{\mathbb{k}}{\otimes}_{\mathbb{Z}}{\U0001d524}_{\mathbb{Z}}$ and
${\U0001d524}_{\u2102}=\u2102{\otimes}_{\mathbb{Z}}{\U0001d524}_{\mathbb{Z}}$
– and this ${\U0001d524}_{\u2102}$ is simple of Cartan type. Moreover, we assume that the classical subgroup ${\mathbf{G}}_{\mathrm{ev}}$ of $\mathbf{G}$ has a $\mathrm{\mathbb{k}}$-split maximal torus. In short, we say that *
$\mathbf{G}$ is a gs-split, $\mathrm{\mathbb{k}}$-split supergroup of Cartan type*. The group of characters of any $\mathrm{\mathbb{k}}$-split maximal torus in ${\mathbf{G}}_{\mathrm{ev}}$, call it Λ, contains the root lattice, since ${\mathbf{G}}_{\mathrm{ev}}$ acts on $\U0001d524$ by the adjoint action: hence Λ is an intermediate lattice lying between ${L}_{r}$ and ${L}_{w}$ (notation of
Section 1).

Now, for any pair $({\U0001d524}_{\u2102},\mathrm{\Lambda})$ as above there exists a Chevalley $\mathrm{\mathbb{k}}$-supergroup ${\mathbf{G}}_{V}$ of Cartan type whose associated pair is exactly $({\U0001d524}_{\u2102},\mathrm{\Lambda})$ – cf. Section 1 above. This yields an “Existence Theorem” for gs-split, $\mathrm{\mathbb{k}}$-split supergroups of Cartan type.

A related “Uniqueness Theorem”
was presented in [5, Section 4.8]. However, it was based on a wrong analysis, so it requires
crucial amendments. In fact, I shall present a double version of such a result, one holding true for any ring – but requiring a stronger assumption than that concerning $Lie(\mathbf{G})$ – and one that applies only for fields of zero characteristic – for which the above requirement on $Lie(\mathbf{G})$ is enough.

Here comes the first result:

*Let $\mathrm{G}$ be a connected gs-split, $\mathrm{\mathbb{k}}$-split supergroup of Cartan type, let $\mathrm{(}{\mathrm{g}}_{\mathrm{C}}\mathrm{,}\mathrm{\Lambda}\mathrm{)}$ be its associated pair and let ${\mathrm{G}}_{V}$ be the Chevalley supergroup of Cartan type whose group is associated with the same pair $\mathrm{(}{\mathrm{g}}_{\mathrm{C}}\mathrm{,}\mathrm{\Lambda}\mathrm{)}$.
Assume in addition that ${\mathrm{G}}_{\mathrm{ev}}$ is isomorphic to ${\mathrm{(}{\mathrm{G}}_{V}\mathrm{)}}_{\mathrm{ev}}$.
Then $\mathrm{G}$ is isomorphic to ${\mathrm{G}}_{V}$.*

#### Proof.

To begin with, recall that the classical subgroup ${({\mathbf{G}}_{V})}_{\mathrm{ev}}$ of the supergroup ${\mathbf{G}}_{V}$ splits into a semidirect product ${\mathbf{G}}_{V}={\mathbf{G}}_{0}^{V}\u22c9{\mathbf{G}}_{{\overline{0}}^{\uparrow}}^{V}$, where ${\mathbf{G}}_{0}^{V}$ is a connected $\mathrm{\mathbb{k}}$-split
subgroup with $Lie({\mathbf{G}}_{0}^{V})={\U0001d524}_{0}$ and group of characters Λ, and ${\mathbf{G}}_{{\overline{0}}^{\uparrow}}^{V}$ is a connected, unipotent normal subgroup with $Lie({\mathbf{G}}_{{\overline{0}}^{\uparrow}}^{V})={\U0001d524}_{{\overline{0}}^{\uparrow}}$. In addition, by construction ${\mathbf{G}}_{0}^{V}$ is obtained via a classical procedure “à la Chevalley” – cf. [5, Proposition 4.9] – based on *V* thought of as a ${\U0001d524}_{0}$-module, the Kostant form of $U({\U0001d524}_{0})$, etc. Similarly, ${\mathbf{G}}_{{\overline{0}}^{\uparrow}}^{V}$ too is realized via a construction “à la Chevalley”, which roughly speaking “integrates” the nilpotent Lie algebra ${\U0001d524}_{{\overline{0}}^{\uparrow}}$ (linearized through *V*). These Chevalley constructions for ${\mathbf{G}}_{0}^{V}$ and ${\mathbf{G}}_{{\overline{0}}^{\uparrow}}^{V}$ are realized simultaneously as parts of the overarching procedure which constructs *all* of ${\mathbf{G}}_{V}$.
In addition (as a consequence), the adjoint action of ${({\mathbf{G}}_{V})}_{\mathrm{ev}}$ onto $Lie({\mathbf{G}}_{V})=\U0001d524$ is uniquely determined by the adjoint action of $Lie({({\mathbf{G}}_{V})}_{\mathrm{ev}})={\U0001d524}_{\overline{0}}$ onto $Lie({\mathbf{G}}_{V})=\U0001d524$
as well as by
Λ.

From the isomorphism $\varphi :{\mathbf{G}}_{\mathrm{ev}}\cong {({\mathbf{G}}_{V})}_{\mathrm{ev}}$ we get a corresponding decomposition of ${\mathbf{G}}_{\mathrm{ev}}$ as a semidirect product,
and also that the adjoint action of ${\mathbf{G}}_{\mathrm{ev}}$ onto $Lie(\mathbf{G})=\U0001d524$ is uniquely determined by the action of $Lie({\mathbf{G}}_{\mathrm{ev}})={\U0001d524}_{\overline{0}}$ onto $Lie(\mathbf{G})=\U0001d524$ and by Λ.

Putting all this together we get, in the language of [7], that one can express this by saying that the “super Harish-Chandra pairs” of ${\mathbf{G}}_{V}$ and $\mathbf{G}$, namely $({({\mathbf{G}}_{V})}_{\mathrm{ev}},\U0001d524)$ and $({\mathbf{G}}_{\mathrm{ev}},\U0001d524)$, are isomorphic. Now, the main result in [7] is exactly – cf. Theorem 4.3.14 therein – that the (suitably defined) category of “super Harish-Chandra pairs” is equivalent to the category of (fine) gs-split supergroups. But the latter category contains both our supergroups $\mathbf{G}$ and ${\mathbf{G}}_{V}$ (by assumption for the former, and by the remark at the beginning of the proof for the latter): therefore, we can conclude that $\mathbf{G}$ is isomorphic to ${\mathbf{G}}_{V}$.
∎

The second result is a direct consequence:

*Let $\mathrm{G}$ be a connected gs-split, $\mathrm{\mathbb{k}}$-split supergroup of Cartan type, let $\mathrm{(}{\mathrm{g}}_{\mathrm{C}}\mathrm{,}\mathrm{\Lambda}\mathrm{)}$ be its associated pair and let ${\mathrm{G}}_{V}$ be the Chevalley supergroup of Cartan type associated with $\mathrm{(}{\mathrm{g}}_{\mathrm{C}}\mathrm{,}\mathrm{\Lambda}\mathrm{)}$. Moreover, assume that $\mathrm{\mathbb{k}}$ is a **field* of characteristic zero.
Then $\mathrm{G}$ is isomorphic to ${\mathrm{G}}_{V}$.

#### Proof.

From the proof of Theorem 3.2 above we know that the subgroup ${({\mathbf{G}}_{V})}_{\mathrm{ev}}$ of ${\mathbf{G}}_{V}$ splits as ${\mathbf{G}}_{V}={\mathbf{G}}_{0}^{V}\u22c9{\mathbf{G}}_{{\overline{0}}^{\uparrow}}^{V}$, where ${\mathbf{G}}_{0}^{V}$ is connected, $\mathrm{\mathbb{k}}$-split reductive with $Lie({\mathbf{G}}_{0}^{V})={\U0001d524}_{0}$ and group of characters Λ, and ${\mathbf{G}}_{{\overline{0}}^{\uparrow}}^{V}$ is connected, unipotent, normal with $Lie({\mathbf{G}}_{{\overline{0}}^{\uparrow}}^{V})={\U0001d524}_{{\overline{0}}^{\uparrow}}$. Moreover, the (conjugacy) action of ${\mathbf{G}}_{0}^{V}$ onto ${\mathbf{G}}_{{\overline{0}}^{\uparrow}}^{V}$ is entirely encoded by the adjoint action of ${\U0001d524}_{0}$ onto ${\U0001d524}_{{\overline{0}}^{\uparrow}}$ and by the lattice Λ.

On the other hand, the classical subgroup ${\mathbf{G}}_{\mathrm{ev}}$ of $\mathbf{G}$ has tangent Lie algebra $Lie({\mathbf{G}}_{\mathrm{ev}})={\U0001d524}_{\overline{0}}={\U0001d524}_{0}\oplus {\U0001d524}_{{\overline{0}}^{\uparrow}}$, where ${\U0001d524}_{0}$ is a reductive Lie subalgebra and ${\U0001d524}_{{\overline{0}}^{\uparrow}}$ is a nilpotent ideal of ${\U0001d524}_{\overline{0}}$ (see the proof of Lemma 1.1). Since $\mathrm{\mathbb{k}}$ is a field of characteristic zero, we have a Chevalley decomposition of ${\mathbf{G}}_{\mathrm{ev}}$ into a semidirect product ${\mathbf{G}}_{\mathrm{ev}}={\mathbf{G}}_{0}\u22c9{R}_{u}({\mathbf{G}}_{\overline{0}})$, where ${R}_{u}({\mathbf{G}}_{\overline{0}})$ is the unipotent radical of ${\mathbf{G}}_{\overline{0}}$; then this ${R}_{u}({\mathbf{G}}_{\overline{0}})$ is a connected, unipotent normal subgroup with $Lie({R}_{u}({\mathbf{G}}_{\overline{0}}))={\U0001d524}_{{\overline{0}}^{\uparrow}}$, while ${\mathbf{G}}_{0}\phantom{\rule{veryverythickmathspace}{0ex}}(\cong \mathbf{G}/{R}_{u}({\mathbf{G}}_{\overline{0}}))$ is a connected, $\mathrm{\mathbb{k}}$-split reductive subgroup with $Lie({\mathbf{G}}_{0})={\U0001d524}_{0}$ and group of characters the lattice Λ.

By classification theory of $\mathrm{\mathbb{k}}$-split reductive groups,
${\mathbf{G}}_{0}$ corresponds to the pair $({\U0001d524}_{0},\mathrm{\Lambda})$; but ${\mathbf{G}}_{0}^{V}$, which also is $\mathrm{\mathbb{k}}$-split reductive, corresponds to the same pair as well, whence there exists an isomorphism ${\mathbf{G}}_{0}\cong {\mathbf{G}}_{0}^{V}$. Similarly, in the classification of connected unipotent (algebraic) group-schemes over fields of characteristic zero both ${\mathbf{G}}_{{\overline{0}}^{\uparrow}}^{V}$ and ${R}_{u}({\mathbf{G}}_{\overline{0}})$ correspond to the same nilpotent Lie algebra (namely ${\U0001d524}_{{\overline{0}}^{\uparrow}}$), hence there exists an isomorphism between them. In addition, the (conjugacy) action of ${\mathbf{G}}_{0}$ onto ${R}_{u}({\mathbf{G}}_{\overline{0}})$ is again entirely encoded by the adjoint action of ${\U0001d524}_{0}$ onto ${\U0001d524}_{{\overline{0}}^{\uparrow}}$ and by the lattice Λ.

Comparing now ${({\mathbf{G}}_{V})}_{\mathrm{ev}}$ and ${\mathbf{G}}_{\mathrm{ev}}$ we conclude that both are semidirect products, with pairwise isomorphic factors, and the action of the reductive factor onto the unipotent (normal) one is ruled in the same way; so these semidirect products are isomorphic, i.e. ${\mathbf{G}}_{\mathrm{ev}}\cong {({\mathbf{G}}_{V})}_{\mathrm{ev}}$. Then Theorem 3.2 applies, and we find that $\mathbf{G}$ is isomorphic to ${\mathbf{G}}_{V}$.
∎

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