If you are in the United States with an L-1 visa (a temporary visa for intracompany transferees), applying for U.S. lawful permanent residence (a green card) should be simpler than for most other temporary (nonimmigrant) visa holders, as discussed here.
There's no automatic stepping stone from L-1 status to a green card. You still must separately show eligibility for U.S. lawful permanent residence. You might do so based on employment (most likely through your current employer, if it is willing to sponsor you in one of the employment-based green card categories); or through a qualifying family relationship to a U.S. citizen or permanent resident; or in some other category of U.S. immigration law.
Assuming you do have a basis for green card eligibility, you will enjoy several advantages over other types of applicants, including:
We'll elaborate on these points below.
By way of background, most applicants for nonimmigrant visas must prove, as a condition of getting the visa, that they are not ultimately hoping to get a U.S. green card. Their sole intent must be to come to the U.S. on the nonimmigrant visa, maintain their status under that visa, and then leave the U.S. by the expected and allowed time. But with an L-1 visa, you are allowed to simultaneously intend to spend time as a nonimmigrant on your L-1 visa while also pursuing the possibility of a U.S. green card.
Given your professional qualifications, your best bet in applying for green card might be to use the EB-1C category, which means employment-based first preference, a type of "priority worker." The EB-1 category specifically covers multinational executives and managers.
Your U.S. employer will need to apply on your behalf. You will need to have worked as an executive or manager in a qualifying company for at least one out of the three years before your arrival in the United States, and to be taking a similar position with a U.S. branch, affiliate, or subsidiary of the same company.
Fortunately, your employer will not need to undertake the "PERM" or labor certification process on your behalf to help you obtain an EB-1C green card. Labor certification is an expensive, laborious, and lengthy process, in which the employer must attempt to recruit U.S. workers and establish that none of them are qualified, available, and willing to take your job.
Instead, your green card application process will start with your employer filing a visa petition on your behalf, on USCIS Form I-140. The petition will need to be accompanied by several documents, including proof of your L-1 approval, proof that you've worked for the appropriate amount of time as an executive or manager, a description of your job duties, the company's financial statements and recent tax returns, and so on.
Once the visa petition has been approved, you'll be able to submit your application for a green card, or to "adjust status." After several months, you'll be scheduled for an interview, and hopefully be approved as a U.S. permanent resident.
If you elect to hire a lawyer to represent you and handle the green card application process; not a bad idea if you think there might be complications; you can expect to pay a few thousand dollars.