Total positivity and symmetric spaces

Technical Summary: Total Positivity and Symmetric Spaces

Problem Statement
The paper addresses the extension of the theory of total positivity, originally developed by Lusztig for split connected reductive groups and flag varieties, to the setting of symmetric spaces $S = G/K$. While Lusztig's theory relies on canonical bases and specific pinning data (fundamental pairs), the author seeks to define a notion of total positivity for symmetric spaces that is compatible with the natural Poisson geometry and admits a rich cell structure analogous to the double Bruhat cells found in the group case. A specific challenge is establishing a cell decomposition, explicit positive parametrizations, and closure relations for the totally nonnegative part of the symmetric space, $S_{\ge 0}$, without relying on the canonical basis machinery used in the group case.

Methodology
The author employs a geometric and combinatorial approach centered on the embedding of the symmetric space into the group.

1. Embedding and Stratification: The symmetric space $G/K$ is realized as a closed subvariety of $G$ via the map $\iota: gK \mapsto g\theta(g)^{-1}$, where $\theta$ is the involution defining $K$. The author introduces "double Bruhat cells" for the symmetric space, defined as the strata $\mathring{S}_u = \iota^{-1}(B_+ \dot{u} B_- \dot{w}_\bullet)$ for $u \le w_\bullet$ in the Weyl group. These strata are motivated by Poisson geometry, specifically identified as $T$-leaves for the natural Poisson structure on $G/K$.
2. Pinning and Involution: The theory is developed for a specific class of symmetric pairs where the involution $\theta$ is normalized using an $\imath$-root datum and a split pair $(B_+, T)$. The paper restricts to cases where $\theta$ is compatible with the diagram involution (specifically where parameters $\zeta_i = -1$), excluding certain types (e.g., $A_{III}$ with odd parameters, $A_{IV}$ with odd $n$, $E_{III}$).
3. Parametrization: The author utilizes Marsh–Rietsch positive parametrizations for Richardson strata in partial flag varieties. By relating the indexing set of the symmetric space strata to the indexing set of projected Richardson strata via a poset isomorphism, the author constructs explicit positive parametrizations for the cells in $S_{\ge 0}$.
4. Rank-One Calculations: A crucial technical step involves explicit rank-one calculations (Section 3.3) to establish exchange relations between positive root subgroups and negative root subgroups within the symmetric space context. These calculations verify that the transition between different charts involves subtraction-free rational functions, even when square roots appear in intermediate steps (which vanish upon passing to the quotient torus).

Key Contributions and Results
The paper establishes the following main results (Theorem 3.5):

1. Geometric Containment: The totally nonnegative symmetric space $S_{\ge 0}$ is entirely contained in the intersection of the open $B_+$-orbit and the open $B_-$-orbit of $G/K$.
2. Cell Decomposition: $S_{\ge 0}$ admits a decomposition into cells $S^>_{u}$ indexed by $u \in Q = \{u \in W \mid u \le w_\bullet\}$. This decomposition is isomorphic to a product of a Marsh–Rietsch positive set and a totally positive torus $T^{>0}$.
3. Explicit Parametrizations: For each cell, there are explicit isomorphisms of semi-algebraic varieties:
   • $G^>_{v_+, w} \times T^{>0} \xrightarrow{\sim} S^>_{\phi(v,w)}$
   • $H^>_{v_+, w} \times T^{>0} \xrightarrow{\sim} S^>_{\phi(v,w)^{-1}}$
     where $G^>_{v_+, w}$ and $H^>_{v_+, w}$ are positive sets derived from Marsh–Rietsch coordinates and their Chevalley involution images, respectively.
4. Closure Relations: The closure of a cell $S^>_{u}$ is the union of cells $S^>_{u'}$ for $u \le u'$.
5. Subtraction-Free Transitions: The transition maps between the two natural families of parametrizations (associated with $B_+$ and $B_-$) are subtraction-free (i.e., they involve only addition, multiplication, and division of positive real numbers, with no subtraction). This property holds without taking square roots after passing to the quotient torus.
6. Topological Properties: The totally nonnegative symmetric space $S_{\ge 0}$ is contractible. Furthermore, each cell $S^>_{u}$ is a connected component of the real locus of the corresponding double Bruhat cell $\mathring{S}_u(\mathbb{R})$.

Significance and Claims
The paper claims to provide a self-contained theory of total positivity for symmetric spaces that parallels Lusztig's theory for groups but utilizes a different foundational setup.

• Alternative Framework: Unlike Lusztig's approach which uses fundamental pairs and canonical bases, this work uses split pairs and Poisson-geometric motivations (double Bruhat cells as $T$-leaves).
• Simplicity: The author asserts that this approach leads to a "particularly simple cell decomposition, closure order, and subtraction-free transition functions."
• Independence from Canonical Bases: The paper provides a new proof of the core geometric structure of the totally nonnegative part of the group $G_{\ge 0}$ (specifically its containment in open orbits) that bypasses Lusztig's theory of canonical bases, relying instead on the rank-one exchange relations and regular functions.
• Combinatorial Control: The stratification of the symmetric space is shown to be controlled by the same combinatorics (projected Richardson strata) that governs the partial flag varieties, mediated by a poset isomorphism $\phi: P \to Q^{opp}$.

The work does not propose new experimental applications or future directions beyond the immediate mathematical framework of total positivity, cluster structures, and Poisson geometry. It positions itself as a foundational extension of existing theories to the symmetric space setting.