On the intertwining differential operators from a line bundle to a vector bundle over the real projective space

To state the main problems of this paper, we first introduce some notation. Let $G$ be a real reductive Lie group with complexified Lie algebra ${\mathfrak{g}}$. Let $P$ be a parabolic subgroup with Langlands decomposition $P=MAN_{+}$. We write $\mathrm{Irr}(M)_{\mathrm{fin}}$ for the set of equivalence classes of finite-dimensional irreducible representations of $M$. Likewise, let $\mathrm{Irr}(A)$ denote the set of characters of $A$. Then, for the outer tensor product $\varpi\boxtimes\nu\boxtimes\mathrm{triv}$ of $\varpi\in\mathrm{Irr}(M)_{\mathrm{fin}}$, $\nu\in\mathrm{Irr}(A)$, and the trivial representation $\mathrm{triv}$ of $N_{+}$, we put

for an (unnormalized) parabolically induced representation of $G$. Let $\mathrm{Diff}_{G}(I(\sigma,\lambda),I(\varpi,\nu))$ denote the space of $G$-intertwining differential operators $\mathcal{D}\colon I(\sigma,\lambda)\to I(\varpi,\nu)$.

There are two main problems in this paper. The first problem concerns the classification and construction of intertwining differential operators $\mathcal{D}\in\mathrm{Diff}_{G}(I(\sigma,\lambda),I(\varpi,\nu))$ as follows.

Let $K$ be a maximal compact subgroup of $G$. For $\mathcal{D}\in\mathrm{Diff}_{G}(I(\sigma,\lambda),I(\varpi,\nu))$, the kernel $\mathrm{Ker}(\mathcal{D})$ and image $\mathrm{Im}(\mathcal{D})$ are $G$-invariant subspaces of $I(\sigma,\lambda)$ and $I(\varpi,\nu)$, respectively. Let $\mathrm{Ker}(\mathcal{D})_{K}$ and $\mathrm{Im}(\mathcal{D})_{K}$ denote the space of $K$-finite vectors of $\mathrm{Ker}(\mathcal{D})$ and $\mathrm{Im}(\mathcal{D})$, respectively. The following is the other main problem of this paper.

For instance, if $G=SL(3,\mathbb{R})$, then the maximal compact subgroup $K$ is $K=SO(3)$. Problem B then asks how $\mathrm{Ker}(\mathcal{D})_{K}$ and $\mathrm{Im}(\mathcal{D})_{K}$ decompose as $SO(3)$-modules. We shall answer this question in Corollary 6.7 for $SL(n,\mathbb{R})$ with $n\geq 3$.

Problem A for the first order intertwining differential operators for general $(G,P)$ was done around 2000 independently by Ørsted [63], Slovák–Souček [66], and Johnson–Korányi–Reimann [29], which generalize the work of Fegan [20] for $G=SO(n,1)$. See also the works of Korányi–Reimann [49] and Xiao [72] as relevant works for this case. The higher order case is still a work in progress. For some recent studies, see, for instance, Barchini–Kable–Zierau [2, 3], Kable [32, 33, 34, 35, 36, 37], Kobayashi–Ørsted–Somberg–Souček [47], and the first author [54, 55].

Problem A has been paid attention especially in conformal geometry. The Yamabe operator, also known as the conformal Laplacian, is a classical example of a conformally covariant differential operator (cf. [46, 59]). In this paper we consider the projective structure in parabolic geometry. In other words, our aim is to classify and construct “projectively covariant” differential operators.

On the study of intertwining differential operators, the BGG sequence is an important background (cf. [8, 11]). The kernel of the first BGG operators has also been studied carefully from a geometric point of view (cf. [9, 10, 22]). Intertwining differential operators are also studied intensively by Dobrev in quantum physics (cf. [14, 15, 16, 17, 18, 19]).

We next describe the concrete setup of this paper.

In this paper we consider Problems A and B for $(G,P;\sigma)=(SL(n,\mathbb{R}),P_{1,n-1};(\pm,\mathrm{triv}))$, where $P_{1,n-1}$ is the maximal parabolic subgroup of $G$ corresponding to the partition $n=1+(n-1)$ so that $G/P_{1,n-1}$ is diffeomorphic to the real projective space $\mathbb{R}\mathbb{P}^{n-1}$. The representation $(\pm,\mathrm{triv})\in\mathrm{Irr}(M)_{\mathrm{fin}}$ denotes a one-dimensional representation of $M\simeq SL^{\pm}(n-1,\mathbb{R})$. The details will be discussed in Section 3.1. We shall solve Problem A in Theorems 4.2 and 4.5, and Problem B in Corollary 6.7.

In 1949, Bol from projective differential geometry showed that $SL(2,\mathbb{R})$-intertwining differential operators for an equivariant line bundle over $S^{1}$ are only the powers of the standard derivative $\frac{d^{k}}{dx^{k}}$ (cf. [5] and [64, Thm. 2.1.2]). The operator $\frac{d^{k}}{dx^{k}}$ is sometimes called the Bol operator. For recent results on analogues of the Bol operator for Lie superalgebras, see, for instance, [6, 7]. We successfully generalize the classical result of Bol on $S^{1}$, which is a double cover of $\mathbb{R}\mathbb{P}^{1}$, to $SL(n,\mathbb{R})$ on $\mathbb{R}\mathbb{P}^{n-1}$.

It is classically known that the Bol operators $\frac{d^{k}}{dx^{k}}$ are the residue operators of the Knapp–Stein operator. In contrast to the case of $n=2$, the Knapp–Stein operator does not exist for $(SL(n,\mathbb{R}),P_{1,n-1})$ for $n\geq 3$ (cf. [62]). Thus, the differential operators obtained in Theorem 4.5 are not the residue operators of such. In the sense of Kobayashi–Speh [48], our differential operators are all sporadic operators for $n\geq 3$.

Our main tool to work on Problem A is the so-called F-method. This is a fascinating method invented by Toshiyuki Kobayashi around 2010 in the course of the study of his branching program (see, for instance, [40], [47, Introduction], and [69, Sect. B]). Since then, Problem A has been intensively studied by the F-method especially in symmetry breaking setting (cf. [21, 38, 39, 42, 44, 45, 47, 51, 52, 53, 61, 65, 67]).

The F-method makes it possible to classify and construct intertwining differential operators (or more generally speaking, differential symmetry breaking operators) by solving a certain system of partial differential equations. To describe the main idea more precisely, recall that it follows from a fundamental work of Kostant [50] that the space $\mathrm{Diff}_{G}(I(\sigma,\lambda),I(\varpi,\nu))$ of intertwining differential operators is isomorphic to the space of $({\mathfrak{g}},P)$-homomorphisms between generalized Verma modules (see also [12, 14, 23, 27, 49]). Schematically, we have

The precise statement will be given in Theorem 2.5. In general, it is easier to work with the Verma module side “$\operatorname{Hom}(\text{Verma})$” than the differential operator side $\mathrm{Diff}_{G}(I(\sigma,\lambda),I(\varpi,\nu))$. Thus, the standard strategy to tackle Problem A is to convert the problem into the one for generalized Verma modules. Nonetheless, even in a case that the unipotent radical $N_{+}$ is abelian, it requires involved combinatorial computations (see, for instance, [30, Chap. 5]).

The novel idea of the F-method is to further identify $\mathrm{Diff}_{G}(I(\sigma,\lambda),I(\varpi,\nu))$ with the space of polynomial solutions to a system of PDEs by applying a Fourier transform to a generalized Verma module. In other words, the F-method puts another picture “$\textnormal{Sol}(\text{PDE})$” to (1.1) as follows.

Then, in the F-method, one achieves the classification and construction of intertwining differential operators simultaneously by solving the system of PDEs. We shall explain more details in Section 2.

As observed above, one can obtain $({\mathfrak{g}},P)$-homomorphisms in “$\textnormal{Hom}(\text{Verma})$” from the solutions in “$\textnormal{Sol}(\text{PDE})$”. We thus study the algebraic counterpart of Problem A for $({\mathfrak{g}},P)=(\mathfrak{sl}(n,\mathbb{C}),P_{1,n-1})$. It is achieved in Theorems 4.6 and 4.8. We further consider a variant of Problem A for ${\mathfrak{g}}$-homomorphisms between generalized Verma modules in Theorem 5.23. We also classify in Corollary 5.24 the reducibility of the generalized Verma module in consideration; this recovers a result of [1, 24, 25] for the pair $({\mathfrak{g}},{\mathfrak{p}})=(\mathfrak{sl}(n,\mathbb{C}),{\mathfrak{p}}_{1,n-1})$.

A ${\mathfrak{g}}$-homomorphism between generalized Verma modules is called a standard map if it is induced from a ${\mathfrak{g}}$-homomorphism between the corresponding (full) Verma modules; otherwise, it is called a non-standard map ([4, 60]). In this paper we also show that the resulting ${\mathfrak{g}}$-homomorphisms in Theorem 5.23 are all standard.

Kable [31] and the authors [58] recently showed a Peter–Weyl type theorem for the kernel $\mathrm{Ker}(\mathcal{D})$ of an intertwining differential operator $\mathcal{D}$. The theorem allows us to compute the $K$-type formula of $\mathrm{Ker}(\mathcal{D})$ explicitly by solving the hypergeometric/Heun differential equation ([57, 58]). Tamori [68] independently used a similar idea to determine the $K$-type formula of $\mathrm{Ker}(\mathcal{D})$ for his study of minimal representations.

The Peter–Weyl type theorem works nicely for first and second order differential operators; nonetheless, it requires a certain amount of computations for higher order cases. Since the differential operators that we obtained in Theorem 4.5 have arbitrary order, we take another approach in this paper.

In 1990, van Dijk–Molchanov [71] and Howe–Lee [26] independently showed among other things that the degenerate principal series representations in consideration have length two and have a unique finite-dimensional irreducible subrepresentation $F$. (See Möllers–Schwarz [62] for recent development of this matter.) Since the $K$-type structures of the induced representations are also known, the determination of the $K$-type formula of $\mathrm{Ker}(\mathcal{D})$ is equivalent to showing $\mathrm{Ker}(\mathcal{D})\neq\{0\}$. As the length is two, it simultaneously determines the $K$-type formula of $\mathrm{Im}(\mathcal{D})$.

In order to show that $\mathrm{Ker}(\mathcal{D})\neq\{0\}$, we show $\mathcal{D}f_{0}=0$ for a lowest weight vector $f_{0}\in F$. We shall discuss the details in Section 6. We remark that one can also show $\mathrm{Ker}(\mathcal{D})\neq\{0\}$ by using the lowest $K$-type in [33] for $n=3$.

Now we outline the rest of this paper. There are seven sections including the introduction. In Section 2, we review a general idea of the F-method. In particular, a recipe of the F-method will be given in Section 2.9. Since we do not find a thorough exposition of the F-method in the English literature (except the original papers of Kobayashi with his collaborators [44, 45, 47]), we decided to give some detailed account. We hope that it will be helpful for a wide range of readers. It is remarked that part of the section is an English translation of the Japanese article [56] of the first author.

Section 3 is for the specialization of the framework discussed in Section 2 to the case $(G,P)=(SL(n,\mathbb{R}),P_{1,n-1})$. In this section we fix some notation and normalizations for the rest of the sections. Then, in Section 4, we give our main results for Problem A for the triple $(G,P;\sigma)=(SL(n,\mathbb{R}),P_{1,n-1};(\pm,\mathrm{triv}))$. These are accomplished in Theorems 4.2 and 4.5. Further, a variant of Problem A for $({\mathfrak{g}},P)$-homomorphisms between generalized Verma modules is also discussed in this section. These are stated in Theorems 4.6 and 4.8. We give proofs of these theorems in Section 5 by following the recipe of the F-method.

We consider Problem B in Section 6. In this section we classify the $K$-type formulas of the kernel $\mathrm{Ker}(\mathcal{D})$ and the image $\mathrm{Im}(\mathcal{D})$ of the intertwining differential operators $\mathcal{D}$ classified in Section 4. These are obtained in Corollary 6.7.

Section 7 is an appendix discussing the standardness of the ${\mathfrak{g}}$-homomorphisms $\varphi$ obtained in Section 4 between generalized Verma modules. We first quickly review the definition of the standard map. Then, by applying a version of Boe’s criterion, we show that the homomorphisms $\varphi$ are all standard maps. This is achieved in Theorem 7.4.

Let $G$ be a real reductive Lie group and $P=MAN_{+}$ a Langlands decomposition of a parabolic subgroup $P$ of $G$. We denote by ${\mathfrak{g}}(\mathbb{R})$ and ${\mathfrak{p}}(\mathbb{R})={\mathfrak{m}}(\mathbb{R})\oplus{\mathfrak{a}}(\mathbb{R})\oplus{\mathfrak{n}}_{+}(\mathbb{R})$ the Lie algebras of $G$ and $P=MAN_{+}$, respectively.

For a real Lie algebra $\mathfrak{y}(\mathbb{R})$, we write $\mathfrak{y}$ and $\mathcal{U}({\mathfrak{y}})$ for its complexification and the universal enveloping algebra of ${\mathfrak{y}}$, respectively. For instance, ${\mathfrak{g}},{\mathfrak{p}},{\mathfrak{m}},{\mathfrak{a}}$, and ${\mathfrak{n}}_{+}$ are the complexifications of ${\mathfrak{g}}(\mathbb{R}),{\mathfrak{p}}(\mathbb{R}),{\mathfrak{m}}(\mathbb{R}),{\mathfrak{a}}(\mathbb{R})$, and ${\mathfrak{n}}_{+}(\mathbb{R})$, respectively.

For $\lambda\in{\mathfrak{a}}^{*}\simeq\operatorname{Hom}_{\mathbb{R}}({\mathfrak{a}}(\mathbb{R}),\mathbb{C})$, we denote by $\mathbb{C}_{\lambda}$ the one-dimensional representation of $A$ defined by $a\mapsto a^{\lambda}:=e^{\lambda(\log a)}$. For a finite-dimensional irreducible representation $(\sigma,V)$ of $M$ and $\lambda\in{\mathfrak{a}}^{*}$, we denote by $\sigma_{\lambda}$ the outer tensor representation $\sigma\boxtimes\mathbb{C}_{\lambda}$. As a representation on $V$, we define $\sigma_{\lambda}\colon ma\mapsto a^{\lambda}\sigma(m)$. By letting $N_{+}$ act trivially, we regard $\sigma_{\lambda}$ as a representation of $P$. Let $\mathcal{V}:=G\times_{P}V\to G/P$ be the $G$-equivariant vector bundle over the real flag variety $G/P$ associated with the representation $(\sigma_{\lambda},V)$ of $P$. We identify the Fréchet space $C^{\infty}(G/P,\mathcal{V})$ of smooth sections with

the space of $P$-invariant smooth functions on $G$. Then, via the left regular representation $L$ of $G$ on $C^{\infty}(G)$, we realize the parabolically induced representation $\pi_{(\sigma,\lambda)}=\mathrm{Ind}_{P}^{G}(\sigma_{\lambda})$ on $C^{\infty}(G/P,\mathcal{V})$. We denote by $R$ the right regular representation of $G$ on $C^{\infty}(G)$.

Similarly, for a finite-dimensional irreducible representation $(\eta_{\nu},W)$ of $MA$, we define an induced representation $\pi_{(\eta,\nu)}=\mathrm{Ind}_{P}^{G}(\eta_{\nu})$ on the space $C^{\infty}(G/P,\mathcal{W})$ of smooth sections for a $G$-equivariant vector bundle $\mathcal{W}:=G\times_{P}W\to G/P$. We write $\mathrm{Diff}_{G}(\mathcal{V},\mathcal{W})$ for the space of intertwining differential operators $\mathcal{D}\colon C^{\infty}(G/P,\mathcal{V})\to C^{\infty}(G/P,\mathcal{W})$.

Let $\mathfrak{g}(\mathbb{R})=\mathfrak{n}_{-}(\mathbb{R})\oplus\mathfrak{m}(\mathbb{R})\oplus\mathfrak{a}(\mathbb{R})\oplus\mathfrak{n}_{+}(\mathbb{R})$ be the Gelfand–Naimark decomposition of $\mathfrak{g}(\mathbb{R})$, and write $N_{-}=\exp({\mathfrak{n}}_{-}(\mathbb{R}))$. We identify $N_{-}$ with the open Bruhat cell $N_{-}P$ of $G/P$ via the embedding $\iota\colon N_{-}\hookrightarrow G/P$, $\bar{n}\mapsto\bar{n}P$. Via the restriction of the vector bundle $\mathcal{V}\to G/P$ to the open Bruhat cell $N_{-}\stackrel{{\scriptstyle\iota}}{{\hookrightarrow}}G/P$, we regard $C^{\infty}(G/P,\mathcal{V})$ as a subspace of $C^{\infty}(N_{-})\otimes V$.

We view intertwining differential operators $\mathcal{D}\colon C^{\infty}(G/P,\mathcal{V})\to C^{\infty}(G/P,\mathcal{W})$ as differential operators $\mathcal{D}^{\prime}\colon C^{\infty}(N_{-})\otimes V\to C^{\infty}(N_{-})\otimes W$ such that the restriction $\mathcal{D}^{\prime}|_{C^{\infty}(G/P,\mathcal{V})}$ to $C^{\infty}(G/P,\mathcal{V})$ is a map $\mathcal{D}^{\prime}|_{C^{\infty}(G/P,\mathcal{V})}\colon C^{\infty}(G/P,\mathcal{V})\to C^{\infty}(G/P,\mathcal{W})$ (see (2.1) below).

To describe the F-method, one needs to introduce the following:

1. (1)

   infinitesimal representation $d\pi_{(\sigma,\lambda)}$ (Section 2.2),

2. (2)

   duality theorem (Section 2.3),

3. (3)

   algebraic Fourier transform $\widehat{\;\;\cdot\;\;}$ of Weyl algebras (Section 2.4),

4. (4)

   Fourier transformed representation $\widehat{d\pi_{(\sigma,\lambda)^{*}}}$ (Section 2.5),

5. (5)

   algebraic Fourier transform $F_{c}$ of generalized Verma modules (Section 2.6).

We start with the representation $d\pi_{(\sigma,\lambda)}$ of ${\mathfrak{g}}$ on $C^{\infty}(N_{-})\otimes V$ derived from the induced representation $\pi_{(\sigma,\lambda)}$ of $G$ on $C^{\infty}(G/P,\mathcal{V})$.

For a representation $\eta$ of $G$, we denote by $d\eta$ the infinitesimal representation of ${\mathfrak{g}}(\mathbb{R})$. For instance, $dL$ and $dR$ denote the infinitesimal representations ${\mathfrak{g}}(\mathbb{R})$ of the left and right regular representations of $G$ on $C^{\infty}(G)$. As usual, we naturally extend representations of ${\mathfrak{g}}(\mathbb{R})$ to ones for its universal enveloping algebra $\mathcal{U}({\mathfrak{g}})$ of its complexification ${\mathfrak{g}}$. The same convention is applied for closed subgroups of $G$.

For $g\in N_{-}MAN_{+}$, we write

where $p_{\pm}(g)\in N_{\pm}$ and $p_{0}(g)\in MA$. Similarly, for $Y\in{\mathfrak{g}}={\mathfrak{n}}_{-}\oplus{\mathfrak{l}}\oplus{\mathfrak{n}}_{+}$ with ${\mathfrak{l}}={\mathfrak{m}}\oplus{\mathfrak{a}}$, we write

where $Y_{{\mathfrak{n}}_{\pm}}\in{\mathfrak{n}}_{\pm}$ and $Y_{\mathfrak{l}}\in{\mathfrak{l}}$.

Let $X\in{\mathfrak{g}}(\mathbb{R})$. Since $N_{-}MAN_{+}$ is an open dense subset of $G$, we have $\exp(tX)\in N_{-}MAN_{+}$ for sufficiently small $t>0$. Thus, we have

which implies

Further, for $\bar{n}\in N_{-}$ and sufficiently small $t>0$, we have

Observe that

Equation (2.2) shows that, for $X\in{\mathfrak{g}}(\mathbb{R})$, the formula $d\pi_{(\sigma,\lambda)}(X)$ on $\iota^{*}(C^{\infty}(G/P,\mathcal{V}))$ can be extended to the whole space $C^{\infty}(N_{-})\otimes V$. By extending the formula complex linearly to ${\mathfrak{g}}$, we have

for $X\in{\mathfrak{g}}$ and $f(\bar{n})\in C^{\infty}(N_{-})\otimes V$.

For a finite-dimensional irreducible representation $(\sigma_{\lambda},V)$ of $MA$, we write $V^{\vee}=\operatorname{Hom}_{\mathbb{C}}(V,\mathbb{C})$ and $((\sigma_{\lambda})^{\vee},V^{\vee})$ for the contragredient representation of $(\sigma_{\lambda},V)$. By letting ${\mathfrak{n}}_{+}$ act on $V^{\vee}$ trivially, we regard the infinitesimal representation $d\sigma^{\vee}\boxtimes\mathbb{C}_{-\lambda}$ of $(\sigma_{\lambda})^{\vee}$ as a ${\mathfrak{p}}$-module. The induced module

is called a generalized Verma module. Via the diagonal action of $P$ on $M_{\mathfrak{p}}(V^{\vee})$, we regard $M_{\mathfrak{p}}(V^{\vee})$ as a $({\mathfrak{g}},P)$-module.

The following theorem is often called the duality theorem. For the proof, see, for instance, [12, 44, 49].

The Weyl algebra $\mathbb{C}[U;z,\tfrac{\partial}{\partial z}]$ naturally acts on the space $\mathcal{D}^{\prime}(U)$ of distributions on $U$. In particular, the action of $\mathbb{C}[U;z,\tfrac{\partial}{\partial z}]$ preserves the subspace $\mathcal{D}^{\prime}_{0}(U)$ of distributions supported at $0$. Let $\mathcal{J}$ be the annihilator of the Dirac delta function $\delta_{0}$, that is, the kernel of the homomorphism

Then $\mathcal{J}$ is the left ideal of $\mathbb{C}[U;z,\tfrac{\partial}{\partial z}]$ generated by the coordinate functions $z_{1},\ldots,z_{n}$. Since the map $\Psi$ is surjective (see, for instance, [70, Thm. 5.5]), it induces the isomorphism

On the other hand, by applying the algebraic Fourier transform $\widehat{\;\;\cdot\;\;}$ in (2.13) to $\mathbb{C}[U;z,\tfrac{\partial}{\partial z}]/\mathcal{J}$, the space $\mathbb{C}[U;z,\tfrac{\partial}{\partial z}]/\mathcal{J}$ is isomorphic to the space $\mathrm{Pol}(U^{\vee})=\mathbb{C}[U^{\vee};\zeta]$ of polynomials on $U^{\vee}$. Thus, we have

It is remarked that the composition $\widehat{\;\;\cdot\;\;}\circ\bar{\Psi}^{-1}$ maps $\mathcal{D}^{\prime}_{0}(U)\ni\delta_{0}\mapsto 1\in\mathrm{Pol}(U^{\vee})$.

For $2\rho\equiv 2\rho({\mathfrak{n}}_{+})=\mathrm{Trace}(\mathrm{ad}|_{{\mathfrak{n}}_{+}})\in\mathfrak{a}^{*}$, we denote by $\mathbb{C}_{2\rho}$ the one-dimensional representation of $P$ defined by $p\mapsto\chi_{2\rho}(p)=\left|\mathrm{det}(\mathrm{Ad}(p)\colon\mathfrak{n}_{+}\to\mathfrak{n}_{+})\right|$. For the contragredient representation $((\sigma_{\lambda})^{\vee},V^{\vee})$ of $(\sigma_{\lambda},V)$, we put $\sigma^{*}_{\lambda}:=\sigma^{\vee}\boxtimes\mathbb{C}_{2\rho-\lambda}$. As for $\sigma_{\lambda}$, we regard $\sigma^{*}_{\lambda}$ as a representation of $P$. Define the induced representation $\pi_{(\sigma,\lambda)^{*}}=\mathrm{Ind}_{P}^{G}(\sigma^{*}_{\lambda})$ on the space $C^{\infty}(G/P,\mathcal{V}^{*})$ of smooth sections for the vector bundle $\mathcal{V}^{*}=G\times_{P}(V^{\vee}\otimes\mathbb{C}_{2\rho})$ associated with $\sigma^{*}_{\lambda}$, which is isomorphic to the tensor bundle of the dual vector bundle $\mathcal{V}^{\vee}=G\times_{P}V^{\vee}$ and the bundle of densities over $G/P$. Then the integration on $G/P$ gives a $G$-invariant non-degenerate bilinear form $\mathrm{Ind}^{G}_{P}(\sigma_{\lambda})\times\mathrm{Ind}^{G}_{P}(\sigma^{*}_{\lambda})\to\mathbb{C}$ for $\mathrm{Ind}^{G}_{P}(\sigma_{\lambda})$ and $\mathrm{Ind}^{G}_{P}(\sigma^{*}_{\lambda})$.