If Louisiana had a typical party primary system, Democratic gubernatorial candidate John Bel Edwards would be talking to Democrats right now, and Republicans David Vitter, Scott Angelle and Jay Dardenne would be making their case to GOP voters.

That’s not the way things work in the open primary system, of course. One dynamic is that, while Democrats face a steep uphill climb to reach the Governor’s Mansion, the party still offers enough votes — 40 percent, give or take — to tip the balance for any of the GOP candidates.

So it’s no surprise that Republican candidates are quietly reaching across the aisle to see if they can peel off some of that support.

What is a bit unexpected is that Vitter, the most rigidly conservative of the bunch and the one who built a Senate re-election campaign around relentless attacks against President Barack Obama, is conducting what many political players in Democrat-rich, majority African-American New Orleans see as the most aggressive outreach.

Vitter’s definitely been showing up in some unexpected places lately.

One such place was the Zulu Social Aid & Pleasure Club’s annual Carnival season ball, where his presence stirred the ire of at least one high- profile Democrat.

Ken Carter, a former assessor in New Orleans and a longtime member of the BOLD political organization, said he was “very surprised” to see Vitter there and none too happy about it. He said he approached Vitter and told him he was “ashamed of the positions he’s taken,” from opposing a higher minimum wage to trying to stand in the way of Loretta Lynch’s confirmation as U.S. attorney general.

He said he was speaking as a constituent who’s backed Republicans in the past — and only for himself, not on behalf of his daughter, state Sen. and Louisiana Democratic Party Chairwoman Karen Carter Peterson. As for his own leanings this year, Carter said it’s premature to have serious conversations with Republicans and that “it’s better to support a viable Democrat.”

Members of Vitter’s Senate office staff recently showed up at a Citywide Tenant Association meeting about providing computer classes for low-income children. Public housing leader Cynthia Wiggins said the staffers offered to set up a meeting with the senator himself.

A Vitter flyer also recently appeared at the community center at Fauborg Lafitte, the mixed-income complex on the site of the old Lafitte public housing development. It highlights three items he’s “working on now”: “Protecting Social Security,” “Defending Louisiana Seniors from Obamacare Cuts,” and “Fighting for Increased Access to Safe, Affordable Drugs.”

The flyer was issued by his office, which allows him to cast it as constituent service rather than campaign activity.

Last week, Vitter’s campaign held one of its closed-door meetings with community leaders in New Orleans, this time on crime. One agenda item centered on sentencing reform, an issue where Democrats and Republicans are starting to find considerable common ground nationwide. New Orleans Mayor Mitch Landrieu skipped the event, but a couple of other prominent city Democrats, Orleans Parish Sheriff Marlin Gusman and District Attorney Leon Cannizzaro, were on the attendee list.

Then there are the letters.

Vitter recently sent out a mass mailing to every elected official in the state, asking for their support and including an endorsement card for them to sign and send it. It’s safe to say that the request caught some Democratic eyes.

Constable Lambert Boissiere Jr., a veteran of the New Orleans City Council, the state Senate and the New Orleans endorsement wars, tossed his in the trash. That doesn’t mean he’s ready to sign on the dotted line for Edwards. Asked where he stands on the race as of now, Boissiere quoted a poster hanging in his office that features a picture of an orangutan.

“Sometimes I sits and thinks, and sometimes I just sits,” Boissiere said. “For now, I’m just sitting.”

Vitter’s two Republican rivals, each of whom have more moderate reputations, are doing their own outreach as well. Dardenne and Angelle also have been spotted at normally Democratic, largely African-American haunts.

Apparently concerned that Angelle might be making inroads, Edwards is running an ad in Angelle’s home base of Acadiana, on radio stations with large black audiences, casting Angelle as Gov. Bobby Jindal’s “right-hand man,” and linking him to Jindal’s hospital privatization drive and budget cuts at Southern University.

Will it work?

Talk to Democratic leaders, and you hear conflicting instincts.

One school of thought is that, given the state’s overall Republican leanings, it makes sense to sit down with a likely winner and try to secure some policy concessions.

The flip side — and the big challenge for Vitter in particular — is that people have records.

“Many Republicans do not campaign hard in the African-American community,” Carter said. “They shouldn’t, because they have not delivered.”

CLARIFICATION: In my March 20 column, I attributed consultant James Carville’s comments to his letter to the editor in the LSU Reveille. Some of his comments came from an interview with The Advocate’s Gregory Roberts.

Stephanie Grace can be contacted at sgrace@theadvocate.com. Read her blog at http://blogs.theadvocate.com/gracenotes.